Parabola, Ellipse, Hyperbola - Best Approach

Best Approach

Conic Section(Parabola, Ellipse, Hyperbola)

Pattern-2

WorkBook

Manoj Chauhan Sir (IIT Delhi)

Exp. More than 13 Yearsin Top Coaching of Kota

No. 1 Faculty of Unacademy,

By Mathematics Wizard

Maths IIT-JEE ‘Best Approach’ Conic SectionMCSIR

2Get 10% Instant Discount On Unacademy Plus [Use Referral Code: MCSIRLIVE]

CONIC SECTION

Locus of a point moves in a plane such that ratio of itsdistance from a fixed point to its distance from a fixed lineis constant.(i) The fixed point is called focus(ii) The fixed line is called directrix(iii) Constant Ratio is called eccentricity (e)(iv) Line passing through focus and perpendicular to

directrix is called axis.(v) Point of intersection of conic with axis is called

vertex.

General equation of conic :-Let (, ) and lx + my + n = 0, then

(l2 + m2 ) {(x – )2 + (y – )2} = e2 (lx + my + n)2

Case 1 ( = 0) :-Focus lies on directrix :-( = abc + 2fgh – af2 – bg2 – ch2 = 0)

1. e > 12. e = 13. (e < 1)

Case 2 ( 0) :Focus does not lies on directrix.(abc + 2fgh – af2 – bg2 – ch2 0)

Parabola : e = 1 and h2 – ab = 0

Ellipse : h2 – ab < 0 0 < e < 1 0

Hyperbola : e > 1 , h2 > ab, 0

Rectangular hyperbola :Special case of hyperbola.

e 2 and 2h ab 0

Q. Find locus of a point which moves such that theRatio of its distance from (1,2) to its perpendicular

distance from 4x – 3y + 2 = 0 is 3 .

Q. Check whether the equation is of parabola orellipse or hyperbola.

(x – 2)2 + (y – 5)2 =

24x 3y 8

5

PARABOLA

Standard equation of parabola and generalTerminology :

Some Important Definitions :

Focal Directrix property :

Focal Distance :

Focal chord :

Double ordinate :

Latus Ractum :

Two parabola are said to equal if they have sameLatus Rectum :

Note(i) Perpendicular distance from focus to directrix =

half the latus rectum.(ii) Vertex is middle point of the focus & the point of

intersection of directrix & axis.(iii) Point of intersection of axis and directrix is called

foot of directrix.(iv) Two parabolas are said to be equal if they have

the same latus rectum.

Four standard forms of the parabola are y² = 4ax,y² = – 4ax, x² = 4ay, x² = – 4ay.

Standard parabola y2 = 4ax (x > 0) at a glance:

Four standard form of parabola.(i) Vertex = (0,0)

foot of directrix (–a, 0)focus (a , 0)

(i)(–a,0) (a,0)(0,0)

y = 4ax2

(–a,0)



(ii)(–a,0) (a,0)

y = –4ax2 focus (–a, 0)

foot of directrix = (a,0)

(iii)

(0,a)

(x= 4ay)2

(0,–a)

focus = (0,a)foot of directrix = (0,–a)

(iv)

(0,a)

(x = – 4ay)2 (0,–a)

focus = (0,–a)foot of directrix = (0,a)

In case the vertex of parabola is (h, k)(i) Axis is | | to x–axis then its equation can be taken

as (y – k)2 = 4a (x – h).(ii) Axis is | | to y–axis then its equation can be taken

as (x – h)2 = 4a (y – k).

Q. Identify that the given equation is parabola ornot : 2x2 + 3xy + 7y2 – 8x – 7y – 1 = 0.

Parametric Representation :(1) x = at2

y = 2atis parametric representation of y2 = 4ax

(2) for x2 = 4ayx = 2aty = at2

Note : ‘a’ is distance from vertex to focus from vertexto foot of directrix.

Q. Find everything for the parabola :4y2 + 12x – 20y + 67 = 0.

Q. Find everything for the parabola :(i) x2 + 2x + 4y = 0(ii) 4x2 + 2y = 8x – 7

Q. Find the equation of parabola :(i) whose focus (1, –1) and directrix x + y – 7 = 0.(ii) whose vertex is (4, –3), latus rectum is 4 and

axis is parallel to the x-axis.

Q. Find equation of parabola passing through(–4, –7), axis is | | to x– axis and vertex at point(4, –3).

Q. If a variable circle touches a fixed circle and afixed line then prove that the locus of the centreof the variable circle is a parabola whose directrixis parallel to a given line at a distance equal to theradius of the given circle.

Q. Find everything about the parabola, which hastangent at vertex 3x – 4y = 5 and focus (1, 2).

Q. Prove that the area of the triangle whose verticesare (x

i , y

i ), i = 1, 2, 3 and inscribed in the parabola

y2 = 4ax is 1

8a|(y

1 – y

2) (y

2 – y

3) (y

3 – y

1) |.

Q. Find the side of an equilateral triangle inscribedin y2 = 8x if one of its vertex coincides with thevertex of the parabola.

Q. A variable circle always passes through(1, 0) and touches the curve y = tan (tan–1x).Find the equation to the locus of its centre.

Q. Find Locus of point of trisection of doubleordinate of parabola y2 = 4ax.

Q. A variable parabola is drawn to pass through A& B, the ends of a diameter of a given circle withcentre at the origin and radius ‘c’ & to have asdirectrix a tangent to a concentric circle of radius‘a’ (a > c); the axes being AB & a perpendiculardiameter, prove that the locus of thefocus of the parabola is the standard ellipse

2 2

2 2

x y

a b = 1 where b2 = a2 – c2



Position of point w.r to parabola :

Chord joining two point :

(–a,0)

chord

(0,0)

t2

t1

chord

meaning of t (at , 2at )2 2 2

2

1

y =4ax2

1 2

1 2 1 2

x 2at ty ....(i)

t t t t2

Special cases :

Note :(i) If the chord passing through point

(c, 0) (Fix point of x-axis or axis on parabola)c = –at

1t2

Focal chord :Chord always passes though focus on parabola.

Q. LOL' and MOM' are two chords of parabolay2 = 4ax with vertex A passing through a point Oon its axis. Prove that the radical axis of the circlesdescribed on LL' and MM' as diameters passesthrough the vertex of the parabola.

Q. A quadrilateral is inscribed in a parabola y2 = 4axand three of its sides pass through fixed points onthe axis. Show that the fourth side also passesthrough fixed point on the axis of the parabola.

Q. A circle and a parabola y2 = 4ax intersect in fourpoints ; show that the algebraic sum of theordinates of the four points is zero.Also show that the line joining one pair of thesefour points and the line joining the other pair areequally inclined to the axis.

Q. All chords of the parabola subtending a right angleat the vertex passes through a fixed point (4a, 0).

Q. Prove that on the axis of any parabola there is acertain point K which has the property that, if achord PQ of the parabola be drawn through it,

then 2 2

1 1

PK QK is the same for all positions

of the chord.

Line and parabola :Let line y = mx + c ... (i)Standard Parabola y2 = 4ax ...(ii)Solve and find D.

Tangent at a point of parabola :yy

1 = 2a(x + x

1) {T = 0}

ty = x + at2

y = mx + a/m

Q. Find equation of circle on AB is a diameter.

y

x

B

A

O

(x2, y )2

(x1, y )1

Q. A circle is described whose centre is the vertexand whose diameter is three quarters of the latusrectum of a parabola, prove that the commonchord of the circle and parabola bisects thedistance between the vertex and the focus.

Length of chord of parabola :Line given y = mx + cparabola given y2 = 4ax

Special Case :

Length of focal chord :

Director circle :Locus of intersection of perpendicular tangents.

Q. A tangent to a parabola y2 = 8x makes an angle45° with the line y = 3x + 5. Find its equation andalso its point of contact.

General Note :Point of intersection of two tangents att1 & t

2 is (at

1t2 , a(t

1 + t

2)).



Q. Find the Ratio of Area of le formed by threepoints on parabola and the le formed by tangentsat those point taken in pair. [IIT]

Normal to the parabola :1. Cartesian form

11 1

yy y x x

2a

2. Slope form :Learn y = –tx + 2at + at3

Normal in slope form ‘m’ :-y = mx – 2am – am3 ....(i)

Note :- If the normal passing through (h,k), then.

Comparison between normal and equation of chord

Normal 3y tx 2at at ....(i)

Equation chord 1 2

1 2 1 2

x at ty

t t t t

2 2

....(ii)

# Let given a parabola y2 = 4ax. Then therelation between t

1 and t

2 : -

# Equation of normal line at parabolay = –t

1x + 2at

1 + at

13

y = –t2x + 2at

2 + at

23

Point of intersection of the normals is :

2 21 2 2 2 1 2 1 2x, y a t t t t 2 , at t t t

Q. If a chord which is normal to the parabolay2 = 4ax at one end subtend a right angle at thevertex, prove that it is inclined at an angle

1tan 2 to the axis and the normal chord

passes through (4a, 0).

Q. Find equation of line touching both the parabolay2 = 4ax and x2 = –32 y.

Q. Let the tangent to the parabola y2 = 4ax meet theaxis in T and the tangent at vertex A in Y. If therectangle TAYG is completed. Find focus of G.

Q. A normal at any point P meets the axis inG and Tangent at vertex in Y if A isvertex and rectangle GAYQ is completed. Showthat equation of locus of Q is x3 = 2ax2 + ay2.

Q. Find the equation of circle passing through focusand touches the parabola y2 = 4ax at point(at2, 2at).

Q. A pair of tangents are drawn which are equallyinclined to a straight line y = mx + c whoseinclination to the axis is , prove that the locus oftheir point of intersection is the straight liney = (x – a) tan 2

Q. If tangents are drawn to y2 = 4ax from any point

P on the parabola y2 = a(x + b) then show that

the normals drawn at their point of contact meet

on a fixed line.

Q. Prove that two parabola y2 = 4ax and

y2 = 4c (x – b) cannot have a common normal

other then the axis unless b

2.a c

In other

words this gives the condition for the two curves

to have a common normal other than x-axis.

Q. If a2 > 8b2 P.T. a point can be formed such that

two tangents drawn form it to the parabola

y2 = 4ax are normals to the parabola x2 = 4by.

Q. Normal to the parabola y2 = 12x at P(3, 6) meet

it again at the point Q. Find the equation of the

circle described on PQ as diameter.

Q. If the normals to the parabola y2 = 4ax at point

(t1 and t

2) intersect again on parabola at point t

3.

Show that t3 = –(t

1 + t

2) also S.T. chord joining

t1, t

2 always passing through a fixed point. Find

the fixed point.

Q. TP and TQ are Tangents to parabola y2 = 4ax

and normals at P and Q. Meet at R on the curve

prove that centre of circle circum scribing le

TPQ lies on parabola. 2y2 = a (x – a).point.

Sub-tangent and Sub-Normal :

Length of sub-tangent at any point P(x,y) on the

parabola y2 = 4ax is twice the abscissa of point

P.



Note :

(i) Subtangent is bisected at vertex

projection of normal

length of normal

sub normal

(a+2, 2at)

att

xy

(at, 0)2

(–at, 0)2

Losubtangent

at –(–at )2

= 2at

2

2

(ii) Length of subnormal is constant for all point on

parabola and is equal to semi latusrectum.

Pair of Tangents :

SS1 = T2

S = equation of curve

S1 = power of point

T = equation of tangent

Chord of contact

Q. Line 3x + y = 6 intersect the parabola y2 = 4x at

A and B. Find co-ordinates of point of intersection

of Tangent drawn at A and B.

Q. Pair of tangent are drawn to parabola y2 = –4x

from every point on the line 3x + y = 2. Prove

that there chord of contact passes through a fix

point.

Length of chord of contact

Area of le PAB formed by pair of tangents and

their chord of contact.

3/22

1y 4ax Area =

2a

Q. From a point on the line x + 4a = 0 pair of tangents

are drawn to the parabola y2 = 4x. Prove that

the chord of contact subtend 90° at vertex.

Chord in terms of mid point :

T = S1

yy1 – 2a (x + x

1) = y

12 – 4ax

1

Q. Find the equation of chord of parabolay2 = 8x, whose mid point is (2, –3).

Q. Find locus of mid-point of chord of parabolay2 = 4ax. Which always passes through focus.

Diametre of parabolaFor all conic : Locus of mid-point of system of| | chords is called diameter.For parabola equation of dimeter is :

2ay

m (m is the slope of chords)

Highlights

(a) Reflection property of parabola. If thetangent and normal at any point ‘P’ of theparabola intersect the axis at ‘T’ and ‘G’ thenST = SG = SP.

(b) Circle circumscribing the le formed by tangent,normal and x-axis has its centre at focus.

(c) Portion of tangent to a parabola cut off betweenthe directrix and the curve subtends 90° at focus.

(d) The tangents at the extremities of a focal chordintersect at right angles on the directrix, and hencea circle on any focal chord as diameter touchesthe directrix. Also a circle on any focal radii of apoint P (at2, 2at) as diameter touches the tangentat the vertex and intercepts a chord of length

a 1 2 t on a normal at the point P..

(e) Any tangent to parabola and perpendicular on itfrom focus meet on tangent at vertex.

Q. Tangent and normal at axtimities or latus ractumor parabola y2 = 4ax from a square.

(f) Semilatus ractum of parabola y2 = 4ax is H.M.between segment of any focal chord of theparabola.

(g) Circle circumscribing the le formed by any threetangents to a parabola passes through the focus.

(h) Orthocentre of any le formed by three tangentsto the parabola lies on the directrix.



Q. A circle circumscribing the triangle formed by

three co-normal points passes through the

vertex of the parabola and its equation is

2(x2 + y2) – 2(h + 2a) x – ky = 0.

Q. Consider a parabola having (1, 2) as its focus

and touching both the coordinate axes. Then

(A) y + 2x = 0 is directrix of parabola

(B) y + x = 2 is tangent at vertex to parabola

(C) length of latus rectum of parabola is 2 2

(D) x2 + 4y2 – 4xy – 10x – 20y + 25 = 0 is the

equation of parabola

Comprehension (1 to 3)

If x + y = 2 and x – y = 0 be the tangent at vertex

and axis of parabola. Straight line 2x – y – 4 = 0

is a tangent to the parabola, then :

1. Focus of the parabola is

(A)1 1

,3 3

(B)2 2

,3 3

(C)4 4

,3 3

(D)(2, 2)

2. Length of latus-ractum of the parabola is

(A) 4 2

3(B) 4 2

(C) 16 2

3(D)

82

3

3. Equation of directrix of the parabola is

(A) 3x + 3y – 4 = 0 (B) 3x + 3y – 2 = 0

(C) 3x + 3y – 8 = 0 (D) x + y – 2 = 0

Q. Let P be a point on the parabola

y2 – 2y – 4x + 5 = 0, such that the tangent on the

parabola at P intersects the directrix at point Q.

Let R be the point that divides the line segment

QP externally in the ratio 1:2

. Find the locus of

R. [JEE 2004, 4 out of 60]

Q. Let A and B be two distinct points on the parabola

y2 = 4x. If the axis of the parabola touches a

circle of radius r having AB as its diameter, then

the slope of the line joining A and B can be

[JEE 2010]

(A) –1/r (B) 1/r

(C) 2/r (D) –2/r

Q. Let S be the focus of the parabola y2 = 8x and let

PQ be the common chord of the circle

x2 + y2 – 2x – 4y = 0 and the given parabola.

The area of the triangle PQS is [JEE 2012]

Q. Let the curve C be the mirror image of the

parabola y2 = 4x, with respect to the line

x + y + 4 = 0. If A and B are the points of

intersection of C with the line y = – 5, then the

distance between A and B is

[IIT JEE Adv. 2015]

Q. Let P be the point on the parabola y2 = 4x which

is at the shortest distance from the center S of the

circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the

point on the circle dividing the line segment SP

internally. Then [IIT JEE Adv. 2016]

(A) SP = 2 5

(B) SQ : QP = ( 5 + 1) : 2

(C) The x-intercept of the normal to the parabola

at P is 6

(D) The slope of the tangent to the circle at Q

is 1

2

Q. If a chord, which is not a tangent, of the parabola

y2 = 16x has the equation 2x + y = p, and midpoint

(h, k), then which of the following is(are) possible

value(s) of p, h and k? [JEE Adv. 2017]

(A) p = – 1, h = 1, k = –3

(B) p = 2, h = 3, k = – 4

(C) p = – 2, h = 2, k = – 4

(D) p = 5, h = 4, k = – 3



ELLIPSE

1. ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is an ellipseif = abc + 2fgh – af2 – bg2 – ch2 0 withh2 < ab0 < e < 1

2. i.e. | PF1 | + | PF2 | = 2a

2 2(x c) y + 2 2(x c) y = 2a (2a > 2c)

3. Standard Equation :-2 2

2 2

x y1

a b (b2 a2–c2)

2 21 22 2

P P1

a b

where P1 is the distance from minor axis and P

2 is

the distance from major axis.

The line containing the two fixed points (called

foci) is called the focal axis and the points ofintersection of the curve with focal axis are calledthe Vertices of the ellipse A1(a, 0) & A2 (– a,0). The distance between F1 & F2 is called theFocal Length.

Distance between the two vertices i.e. 2a is called

the major axis. The distance B1B2 i.e. 2b iscalled the minor axis.

Point of intersection of the major and minor axis

is called the centre of the ellipse. Any chord ofthe ellipse passing through it gets bisected by itand is called the diameter.Major and minor axes together are known asprincipal axes of the ellipse.

Any chord through focus is called a focal chord

and any chord perpendicular to the focal axis iscalled double ordinate.

A particular double ordinate through focus or a

particular focal chord perpendicular to focal axisis called its Latus Rectum.

4. ECCENTRICITY :Definition: Degree of flatness of Ellipse :

2

2

b ce 1

a a

2 2

2 2 2

x y1

a a (1 e )

Note that two ellipse are said to be similiar if theyhave same eccentricity

Length of LR : 22b

a

5. DIRECTRIX AND FOCAL DIRECTRIXPROPERTY :

P(x1, y )1

S2 S1

M2 M1

x = – ae x = ae

PS1 = a – ex1, PS2 = a + ex1then PS1 + PS2 = 2a

Q. Find the equation of the ellipse with its centre(1, 2), focus at (6, 2) and passing through thepoint (4, 6). Also find the directrix.

Q. Find the eccentricity of the ellipse if the length ofits minor axis is equal to its focal length?

Q. A tent is in the form of a semi ellipse, the majoraxis of which coincides with the road level. If thewidth of the road is 10 metres and a man 2m highjust reaches the top when 1 metre from a side ofroad, find the greatest height of the arch.

Q. Find the equation of the straight lines joining the

foci of the ellipse 2 2x y

125 16

to the foci of the

ellipse 2 2x y

124 49

. Also find the area of the

figure formed by the foci of these two ellipse.

Q. Find everything for ellipse

(i)2 2x y

1400 144

,

(ii) 9x2 + 4y2 – 18x – 16y – 11 = 0 ,(iii) 4x2 + 16y2 – 2x – 32y = 12



Q. A rod of length a + b moves in such a way thatboth extremities remains on coordinates. A pointP divides the length in ratio b : a measuring fromx-axis. Find locus of P?

Q. C1 and C2 are two fixed circles with radiir1 and r2 such that C2 is contained in C1.A third circle C moves in such a way that ittouches C1 internally and C2 externally, then findthe find locus of its centre.

Q. Show that the locus of the incentre of the variabletriangle PF2F1 is an ellipse whose eccentricity is

2e

1 e where e is the eccentricity of standard

ellipse when P moves on a standard ellipse.

Q. If the focus, centre and eccentricity ofan ellipse are respectively (3, 4) ; (2, 3) and 1/2find its equation.

Auxiliary circle and eccentric angle :

x2 + y2= a2

P a cos , bsin

Q a cos ,a sin

0 < 2.

x

y'

x'

y

O

N

Q

P

P and Q are corresponding points and is calledthe eccentric angle of the point P.

We have, PN b

costantPQ a b

Hence, if from each point on a circle perpendicularare drawn on a fixed diameter then the locus of apoint P dividing these perpendiculars in a constantratio is an ellipse whose auxiliary circle is theoriginal circle.

Q. Distance of a point on the ellipse 2 2x y

16 2

from its centre is 2. Find the eccentric angle ofthe point P.

Position of a point w.r.t. ellipse :

Line and an ellipse :

Equation of tangent :

(i) Point form: 1 1

2 2

xx yy1

a b

Note that point (x1, y1) on the ellipse

(ii) Paramatic form:

x cos ysin1

a b

at the point (a cos , b sin ) on the ellipse.

(iii) Slope form: y = mx + c

y = mx ± 2 2 2a m b is always a tangent to the

ellipse for all m R.

Note that if a tangent passing through (h, k)

then (h2 – a2)m2 – 2khm + k2 – b2 = 0 ....(1)

Note :(i) Passing through a given point there can be

maximum two tangents.(ii) Equation (1) can be used to determine the locus

of the point of intersection of two tangentsenclosing an angle

(iii) Point of tengency is :

2 2a m b

,c c

, where

2 2 2c a m b

Director circle :

Locus of point of intersection of tangent to

ellipse is director circle of ellipse.

Director circle of the ellipse is x2 + y2 = a2 + b2.

Q. Find tangent to an ellipse 3x2 + 4y2 = 12, parallel

to the line y + 2x = 4. Also find the point of

tengency.

Q. Find equation of the tangent to an ellipse9x2 + 16y2 = 144 passing from (2, 3). Alsocompute the tangents to the ellipse2x2 + 7y2 = 14 from (5, 2)

Q. Tangent to an ellipse makes angles 1, 2 withmajor axis. Find the locus of their intersectionwhen cot 1 + cot 2 = k2.



Q. Two tangents to an ellipse intersect at right angles.

Prove that the sum of the square of the chordswhich the auxiliary circle intersects on them isconstant and equal to the square on the line joiningthe foci.

Q. Locus of the feet of the perpendicular from centreupon a variable tangent to the standard ellipse is(x2 + y2)2 = a2x2 + b2y2.

Q. If s, s' are the lengths of perpendicular on atangent to the ellipse from the foci ; p, p' from thevertex and c that from the centre then show thatc2 – ss' = e2(c2 – pp')

Pair of tangents : SS1 = T2

Note: The tangents at the extremities of the latus rectumof the ellipse intersect at the foot of thecorresponding directrix and the figure formed bythem is a rhombus of area 2a2/e.

POINT OF INTERSECTION OF THE TANGENTSPoint of intersection of the tangents at the point & is

2 2

2 2

cos sina , b

cos cos

Q. Locus of the point of intersection of the pair oftangents on an ellipse if the difference of the pointof eccentric angle of their point of contact is

32 .

Chord of contact :

21

21

b

yy

a

xx = 1 (T = 0)

Normals :

Equation of Normal :

(i) Equation of the normal at (x1 , y1) is

2 2

1 1

a x b y

x y = a2 b2 = a2e2.

(ii) Equation of the normal at the point

(a cos , b sin ) is :

ax · sec – by · cosec =(a2 b2).

Co-normal points :From any point in the plane maximum fournormals can be drawn to the ellipse.Four feet of normals on the ellipse are calledco-normal points.

Q. Find the equations to the normals at the ends ofthe latera recta, and prove that each passes throughan end of the minor axis if e4 + e2 = 1

Q. Find the locus of the feet of the perpendicularsfrom centre on a normal to a standard ellipse.

Q. Find locus of point of intersection of pair oftangents to an ellipse if the sum of the ordinatesof their point of contact is b.

Q. Any ordinate NP of an ellipse meets theauxiliary circle in Q, prove that the locus of theintersection of the normals at P and Q is the circlex2 + y2 = (a + b)2.

Q. Prove that if normal at parametric point, are con current then prove that

sin cos sin 2

sin cos sin 2 0

sin cos sin 2

Equation of chord of an ellipse :The equation of the chord passingthrough the points P(a cos , b sin ) and Q(acos , b sin ) is :

x ycos sin

a 2 b 2

cos

2

If this particular chord passes through (d, 0), then :

d atan tan

2 2 d a

If this particular chord passes through (ae, 0),

then : e 1

tan tan2 2 e 1

Q. If S, H are the foci of an ellipse A is any point onthe curve, ASB, BHC, CSD, DHE ..... chordsand 1, 2, 3, 4 ..... are the eccentric angles ofA, B, C, D .......Prove that

31 2 2tan · tan cot ·cot

2 2 2 2

3 4tan · tan .......2 2



Chord with a given middle point : T = S1

Diameter : The locus of the middle points of asystem of parallel chords with slope 'm' of anellipse is a straight line passing through the centreof the ellipse, called its diameter

y =ma

b2

2

x.

Q. A tangent to the ellipse 2 2

2 2

x y

a b = 1 meets the

ellipse 2 2x y

a b = a + b in the points P and Q.

Prove that the tangents at P and Q are at rightangles.

Q. Tangents are drawn from the point(3, 2) to the ellipse x2 + 4y2 = 9. Find the equationto their chord of contact and the equation of thestraight line joining (3, 2) to the middle point ofthis chord of contact.

Q. Tangents are drawn to the ellipse 2 2

2 2

x y

a b = 1

from the point

22

22

2

ba,ba

a, Prove

that they intercept on the ordinate through thenearer focus a distance equal to the major axis.

Q. Find the locus of the middle points of chords ofan ellipse.

(i) which subtend a right angle at their centre.(ii) the tangent at the ends of which intersect at right

angles.

Q. If Parametric point on ellipse areconcylic P.T. n n I.

H-1 If P be any point on the ellipse with S & S as its

foci then (SP) + (SP) = 2a.

H-2

(i) Product of the length’s of the perpendiculars fromeither focus on a variable tangent to an Ellipse /Hyperbola = (semi minor axis)2 / (semi conjugateaxis)2 = b2

(ii) Feet of the perpendiculars from either foci on avariable tangent to an ellipse / hyperbola lies onits auxiliary circle. Hence deduce that the sum ofthe squares of the chords which the auxiliary circleintercept on two perpendicular tangents to anellipse is constant and is equal to the square onthe line joining the foci.

H-3 If the normal at any point P on the ellipse withcentre C meet the major & minor axes in G & grespectively & if CF be perpendicular upon thisnormal, then

(i) PF . PG = b²(ii) PF . Pg = a²

(iii) PG . Pg = SP . S P(iv) CG . CT = (CS)2

(v) locus of the mid point of Gg is anotherellipse having the same eccentricity as thatof the original ellipse.

H-4 Reflection property :The tangent & normal at apoint P on the ellipse bisect the external & internalangles between the focal distances of P. This refersto the well known reflection property of the ellipsewhich states that rays from one focus are reflectedthrough other focus & viceversa. Hence we candeduce that the straight lines joining each focus tothe foot of the perpendicular from the other focusupon the tangent at any point P meet on the normalPG and bisects it where G is the point where normalat P meets the major axis.

H-5 The portion of the tangent to an ellipse between

the point of contact & the directrix subtends a

right angle at the corresponding focus.

H-6 The circle on any focal distance as diameter

touches the auxiliary circle.

H-7 Perpendiculars from the centre upon all chords

which join the ends of any perpendicular

diameters of the ellipse are of constant length.

H-8 If the tangent at the point P of a standard ellipse

meets the axis in T and t and CY is the

perpendicular on it from the centre then,

(i) Tt . PY = a2 b2 and

(ii) least value of T t is a + b.



HYPERBOLA

GENERAL EQUATION - General 2° curve

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

= abc + 2fgh – af2 – bg2 – ch2 0

h2 > ab

e > 1

Standard Equation and general

terminology :- Standard equation of hyperbola

is deduced using an important property

Difference of distance of point moving in a plane

from two fixed point on same plane is constant.

1 2PF PF 2a

2 2 2 2(x c) y (x c) y 2a (c> a)

2 2

2 2

x y1

a b (c2 a2 + b2)

DEFINITIONS

* Line containing the fixed points F1 and F2 (called

Foci) is called Transverse Axis (TA) or Focal Axis

and the distance between F1 and F2 is called Focal

Length.

* The points of intersection (A1, A2) of the curve

with the transverse axis are called Vertices of the

hyperbola.

* The length '2a' between the vertices is called the

Length of Transverse Axis.

* The perpendicular bisector of Transverse axis is

called the Conjugate Axis (CA). The points B1(0,

b) and B2(0, –b) which have special significance,

are known as the extremities of conjugate axis

and the length '2b' is called the Length of conjugate

axis. The point of intersection of these two axes

is called the centre 'O' of the hyperbola.

(Transverse axis and conjugate axis together are

called the Principal Axis.) Any chord passing

through centre is called Diameter (PQ) and is

bisected by it.

* Any chord passing through focus is called a Focal

Chord

* Any chord perpendicular to the Transverse axis

is called a Double Ordinate (AB).

* A particular double ordinate which passes through

focus or a particular focal chord is perpendicular

to focal axis is called the Latus Rectum (L1L2).

Eccentricity :

(Curvature of hyperbola) e = distance from

centre to focus / distance from focus to vertex.

ce

a

2

2

b1

a or b2 = a2(e2 – 1)

Note:(i) Two hyperbolas are said to be similar if the have

same eccentricity.

(ii) Latus Rectum : 2

2b

a

CONJUGATE HYPERBOLA

For the H : 2

2

2

2

b

y

a

x = 1 ....(1)

CH : 2

2

2

2

b

y

a

x = – 1 ....(2)

Note:(i)(a) If e1 and e2 are the eccentricities of a

hyperbola and its conjugate respectively, then

1ee 22

21

(b) If sec is the eccentricity of a hyperbola then itsconjugate hyperbola has the eccentricity cosec

(ii) The foci of a hyperbola and its conjugate are

concylic and form the vertices of a square.

Directrix and Focal directrix property :Directrix : Corresponding to each focus there are

two lines a

xe

which are known as directrices

and satisfy the focal directrix property ofhyperbola.



D1

M1

x= –

O(–ae,0)F2

P(x , y )1 1

x

D2

M2

F (ae,0)1N1N2

ae x= a

e

y

1 2PF PF 2a

PF1 = ex

1 – a, PF

2 = ex

1 + a

where P (x1, y

1)

Rectangular Hyperbola :If a = b, hyperbola is said to be equilateral orrectangular and has the equation, x2 – y2 = a2.

Eccentricity for such a hyperbola is 2 .

(LR) = 2a(e2 – 1) = 2a = (TA)

Q. Show that locus of centre of circle which touchesexternally two given circles is a hyperbola.

Q. Given the base of a triangle and the ratio of thetangents of half the base angles, prove that thevertex moves on a hyperbola whose foci are theextremities of the base.

Q. An ellipse and a hyperbola are confocal (havethe same focus) and the conjugate axis ofhyperbola is equal to the minor axis of the ellipse.If e

1 and e

2 are the eccentricities of ellipse and

hyperbola then prove that 2 21 2

1 12

e e

Q. Find the equation of hyperbola referred to its

principal axes as the coordinate axes

(a) if the distances of one of its vertices from the foci

are 3 and 1.

(b) whose centre is (1, 0); focus is (6, 0) and

transverse axis 6.

(c) whose centre is (3, 2), one focus is

(5, 2) and one vertex is (4, 2).

(d) whose centre is (–3, 2), one vertex is (–3, 4) and

eccentricity is 5/2.

(e) whose foci are (4, 2) and (8, 2) and eccentricity

is 2.

Q. Find everything for the

H : 9x2 – 18x – 16y2 – 64y + 89 =0

Auxiliary circle and parametric equation :Circle drawn with centre ‘C’ and transverse axisas diametre is Auxiliary circle.Auxiliary circle : x2 + y2 = a2

(for the hyperbola :

2 2

2 2

x y1

a b )

The equations x = a sec & y = b tan together

represents the hyperbola 2 2

2 2

x y1

a b , where

is a parameter.

Position of point w.r. to hyperbola :

Line and hyperbola :

Equation of tangent :(i) Point form : If point on hyperbola (x

1, y

1) then

tangent is 1 12 2

xx yy1

a b

(ii) Parametric form :If point on hyperbola(a sec , b tan ) then tangent is

x sec y tan1

a b

(iii) Slope form : 2 2 2y mx a m b

Note :(i) Point of tengency is :

2 2a m b,

c c

, where ± 2 2 2a m b

(ii) If a tangent passing through (h, k) then

(h2 – a2)m2 – 2khm + k2 + b2 = 0 ....(1)hence passing through a given point there can bemaximum two tangents.



Point of intersection of tangents at A () and B () :

cos sin2 2a ,b

cos cos2 2

DIRECTOR CIRCLE

Locus of intersection of perpendicular tangent to

the hyperbola is director circle of hyperbola.

Director circle of the hyperbola is :

x2 + y2 = a2 – b2.

Q. Tangents are drawn to hyperbola

x2 – y2 = a2 including an angle of 45° find locus of

there point of intersection ?

Q. Find common tangent of y2 = 8x and 3x2 – y2 = 3.

Also find the point of the contact.

Q. Find tangent line to the hyperbola : 2 2x y

136 9

,

passing through point (0, 4)

Q. P.T. two tangents drawn from any point on

hyperbola x2 – y2 = a2 – b2 to the ellipse

2 2

2 2

x y1

a b make complementary angle with

the axis.

Pair of tangent :

SS1 = T2

Chord of hyperbola :

x ycos sin cos

a 2 b 2 2

If this particular chord passes through (d, 0),

then : a dtan tan

2 2 a d

If this particular chord passes through (ae, 0),

then : 1 e

tan tan2 2 1 e

.

Normal :

(i) Normal at (x1, y

1) is

2 22 2

1 1

a x b ya b

x y = a2 e2.

(ii) At (a sec , b tan ) is 2 2ax by

a esec tan

.

Q. Find the equation and length of common tangent

of hyperbola’s. 2 2

2 2

x y1

a b and

2 2

2 2

y x1

a b .

Q. Prove that the part of the tangent at any point of

the hyperbola 2 2

2 2

x y1

a b intercepted between

the point of contact and the transverse axis is a

harmonic mean between the lengths of the

perpendicular drawn from the foci on the normal

at the same point.

Q. If a chord joining the points P (a sec ,

a tan ) & Q (a sec , a tan ) on the hyperbola

x2 – y2 = a2 is a normal to it at P, then show that

tan = tan (4 sec2 – 1)

Chord of contact :

1 12 2

xx yy1

a b , (T = 0)

Chord with a middle point :

T = S1

Pole and Polar are to be interpreted as in case of

parabola or ellipse :

Q. From points on the circle x2 + y2 = a2 tangents are

drawn to the hyperbola x2 – y2 = a2, prove that

the locus of the middle points of the chords of

contact is the curve (x2 – y2)2 = a2(x2 + y2)

Q. A point P moves such that the chord of contact

of the pair of tangents from P on the parabola

y2 = 4ax touches the rectangular hyperbola

x2 – y2 = c2. Show that the locus of P is the ellipse

1)a2(

y

c

x2

2

2

2

.



Q. Find the equation to the locus of the middle pointsof the chords of the hyperbola 2x2 – 3y2 = 1,each of which makes an angle of 45° with the x-axis.

Q. A tangent to the hyperbola 1b

y

a

x2

2

2

2

cuts

the ellipse 1b

y

a

x2

2

2

2

at P and Q. Show that

the locus of the mid point of PQ is

2

2

2

2

2

b

y

a

x

= 2

2

2

2

b

y

a

x

Q. Show that the mid points of focal chords of a

hyperbola 2

2

2

2

b

y

a

x = 1 lie on another similar

hyperbola.

HIGHLIGHTS ON TANGENT AND NORMAL

H1 Locus of the feet of the perpendicular drawn from

focus of the hyperbola 1b

y

a

x2

2

2

2

upon any

tangent is its auxiliary circle i.e. x2 + y2 = a2 &

the product of the feet of these perpendiculars is

b2 · (semi C ·A)2

H2 The portion of the tangent between the point of

contact & the directrix subtends a right angle at

the corresponding focus.

H3 The tangent & normal at any point of a hyperbola

bisect the angle between the focal radii. This spells

the reflection property of the hyperbola as "An

incoming light ray " aimed towards one focus is

reflected from the outer surface of the hyperbola

towards the other focus. It follows that if an ellipse

and a hyperbola have the same foci, they cut at

right angles at any of their common point.

Note that the ellipse 1b

y

a

x2

2

2

2

and

the hyperbola 1bk

y

ka

x22

2

22

2

(a > k > b > 0) are confocal and thereforeorthogonal.

H4 The foci of the hyperbola and the points P and Qin which any tangent meets the tangents at thevertices are concyclic with PQ as diameter of thecircle.

ASYMPTOTES

If the length of the perpendicular let fall from a

point on a hyperbola to a straight line tends to

zero as the point on the hyperbola moves to

infinity along the hyperbola, then the straight line

is called the Asymptote of the Hyperbola.

To find the asymptote of the hyperbola

2 2

2 2

x y+ = 1

a b :

0b

y

a

x2

2

2

2

.

Particular Case : Asymptotes of the rectangular

hyperbola.

x2 y2 = a2 are, y = x which are at right

angles.

Note :

(i) Equilateral hyperbola rectangular hyperbola.

(ii) If a hyperbola is equilateral then the conjugate

hyperbola is also equilateral.

(iii) A hyperbola and its conjugate have the same

asymptote.

(iv) The equation of the pair of asymptotes differ the

hyperbola & the conjugate hyperbola by the same

constant only.

(v) The asymptotes pass through the centre of the

hyperbola & the bisectors of the angles between

the asymptotes are the axes of the hyperbola.



(vi) The asymptotes of a hyperbola are the diagonals

of the rectangle formed by the lines drawn through

the extremities of each axis parallel to the other

axis.

(vii) Asymptotes are the tangent to the hyperbola from

the centre.

(viii) A simple method to find the coordinates of the

centre of the hyperbola expressed as a general

equation of degree 2 should be remembered as:

Let f (x , y) = 0 represents a hyperbola.

Find x

f

&

y

f

. Then the point of intersection of

x

f

= 0 &

y

f

= 0 gives the centre of the

hyperbola.

EXAMPLES ON ASYMPTOTES

Q. Find the asymptotes of the hyperbola,

3x2 – 5xy – 2y2 – 5x + 11y – 8 = 0. Also find the

equation of the conjugate hyperbola and their

centre.

Q. Find the equation to the hyperbola whose

asymptotes are the straight lines 2x + 3y + 3 = 0

and 3x + 4y + 5 = 0 and which passes through

the point (1, – 1). Also write the equation to the

conjugate hyperbola and the coordinates of its

centre.

Q. A normal is drawn to the hyperbola 1b

y

a

x2

2

2

2

at P which meets the transverse axis (TA) at G. If

perpendicular from G on the asymptote meets it

at L, show that LP is parallel to conjugate axis.

Q. A transversal cuts the same branch of a hyperbola

x2/a2 y2/b2 = 1 in P, Pandthe asymptotes in

Q, Q.

Prove that : (i) PQ = P'Q' & (ii) PQ' = P'Q

Q. The tangent at any point P of the hyperbola

1b

y

a

x2

2

2

2

meets one of the asymptotes in Q

and L, M are the feet of the perpendiculars from

Q on the axes. Prove that LM passes through P.

HIGHLIGHTS ON ASYMPTOTES

H1 If from any point on the asymptote a straight line

be drawn perpendicular to the transverse axis,

the product of the segments of this line, intercepted

between the point & the curve is always equal to

the square of the semi conjugate axis.

H2 Perpendicular from the foci on either asymptote

meet it in the same points as the corresponding

directrix & the common points of intersection lie

on the auxiliary circle.

H3 The tangent at any point P on a hyperbola

1b

y

a

x2

2

2

2

with centre C, meets the

asymptotes in Q and R and cuts off a CQR of

constant area equal to ab from the asymptotes &

the portion of the tangent intercepted between

the asymptote is bisected at the point of contact.

This implies that locus of the centre of the circle

circumscribing the CQR in case of a rectangular

hyperbola is the hyperbola itself & for a standard

hyperbola the locus would be the curve,

4(a2x2 b2y2) = (a2 + b2)2.

H4 If the angle between the asymptote

of a hyperbola 1b

y

a

x2

2

2

2

is 2 then e = sec.

RECTANGULAR HYPERBOLA

Rectangular hyperbola referred to its asymptotes

as axis of coordinates.

(a) Equation is xy = c2 with parametric representation

x = ct, y = c/t, t R – {0}.



(b) Equation of a chord joining the points P (t1)

& Q(t2) is x + t1 t2 y = c (t1

+ t2) with slope

m = – 21tt

1.

(c) Equation of the tangent at P (x1 , y1) is

2y

y

x

x

11

& at P (t) is t

x+ ty = 2c.

(d) Equation of normal at (x1, y1) :

2 21 1 1 1xx yy x y & at P (t) is

xt3 yt = c (t4 1).

(e) Chord with a given middle point as

(h, k) is kx + hy = 2hk.

Q. Find everything for the rectangular hyperbola

xy = c2.

Q. A rectangular hyperbola xy = c2 circumscribing a

triangle also passes through the orthocentre of

this triangle. If

ii t

c,ct i = 1, 2, 3 be the angular

points P, Q, R then orthocentre is

321

321

ttct,ttt

c.

Q. If a circle and the rectangular hyperbola xy = c2

meet in the four points t1, t2, t3 & t4 , then

(a) t1 t2 t3 t4 = 1

(b) the centre of the mean position of the four points

bisects the distance between the centres of the two

curves.

(c) the centre of the circle through the points t1, t2 &

t3 is :

1 2 3

1 2 3

c 1t t t ,

2 t t t

1 2 3

1 2 3

c 1 1 1t t t

2 t t t

(d) If PQRS are the four points of intersection of

the circle with rectangular hyperbola then

(OP)2 + (OQ)2 + (OR)2 + (OS)2 = 4r2

where r is the radius of circle.

Documents

Parabola, Ellipse, Hyperbola - Best Approach