Circles Key Concepts-1196

7/17/2019 Circles Key Concepts-1196

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MANOJ CHAUHAN SIR(IIT-DELHI)

EX. SR. FACULTY (BANSAL CLASSES)

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ETOOS Academy Pvt. Ltd. : F- N. Ith Iutl e E Eveee Mt

Mh hw N Oe e Jhlw Kt jth 2

KEY CONCEPTS (CIRCLE)

STANDARD RESULTS :

1. EQUATION OF A CIRCLE IN VARIOUS FORM:

(a) The le wth ete h k & u ‘’ h the equt;

h + k = .

(b) The eel equt le + + + + = wth ete :

& u = g f c2 2+ − .

Remember that every second degree equation in x & y in which coefficient of

x2 = coefficient of y2 & there is no xy term always represents a circle.

I + > ⇒ el le.

+ = ⇒ t le.

+ < ⇒ le.

Nte tht the eel equt le t thee t tt & whh e

t the t tht uque le e thuh thee lle t.

(c) The equt le wth

&

t ete :

+ = .Note that this will be the circle of least radius passing through (x

1 , y

1) & (x

2 , y

2).

2. INTERCEPTS MADE BY A CIRCLE ON THE AXES :

The teet e the le + + + + = the -te e e

g c2 − & f c2 − eetvel.

NOTE :

I > ⇒ le ut the t tw tt t.

I = ⇒ le tuhe the -.

I < ⇒ le le letel ve elw the -.

3. POSITION OF A POINT w.r.t. A CIRCLE :

The t

e ute the le + + + + = .

+ +

+

+ ⇔ .

Note : The etet & the let te t le

wth ete & u + & eetvel.

4. LINE & A CIRCLE :

et = e le & = e le. I the u the le & the leth the

eeul the ete the le the :

(i) > ⇔ the le e t eet the le . e. e ut e the le.

(ii) = ⇔ the le tuhe the le.

(iii) < ⇔ the le et the le.

(iv) = ⇒ the le ete the le.

5. PARAMETRIC EQUATIONS OF A CIRCLE :

The et equt h + k = e :

= h + θ ; = k + θ ; π < θ ≤ π whee h k the ete

the u & θ ete.

Nte tht equt tht le j tw t & the le + =

α β+

2 + α β+

2 =

α β−

2.





6. TANGENT & NORMAL :

(a) The equt the tet t the le + = t t t

+

= . Hee equt tet t ;

+ = . The t teet the tet t the t

2

2

cos

cosaβ−α

β+

asin

cos

α β

α β

+

−2

2

.

(b) The equt the tet t the le + + + + = t t t

+

+ +

+ +

+ = .

(c) = + lw tet t the le + = = + the t tt

−

a m

c

a

c

2 2

, .

(d) I le l / thl t le the t ut thuh the ete the le. U

th t l t the le + + + + = t

=

y f

x g

1

1

++

.

7. A FAMILY OF CIRCLES :

(a) The equt the l le thuh the t teet tw le

= &

= :

+ K

= K ≠ .

(b) The equt the l le thuh the t teet le

= & le = ve + K = .

(c) The equt l le thuh tw ve t

&

e wtte

the :

+

+

K

x y

x yx y

1

11

1 1

2 2

= whee K ete.

(d) The equt l le tuh e le =

t the e t

+

+ K [

] = whee K ete.

I e the le thuh

llel t - the equt the l le tuh t

t

ee

+

+ K

= .

l le llel t - the equt the l le tuh t t

ee

+

+ K

= .

(e) Equt le u tle whe e e ve = ;

= &

= ve ;

+ λ

+ µ

= ve -eet = & -eet

= -eet .

(f) Equt le u qultel whe e e e eeete the le

=

=

= &

=

+ λ

= ve -eet

= -eet -eet = .

8. LENGTH OF A TANGENT AND POWER OF A POINT :

The leth tet etel t

t the le

≡ + + + + = ve = x y gx f y c1

2

1

2

1 12 2+ + + + = S1.

que leth the tet the t l lle THE POWER OF POINT w..t. le.

we t e tt w..t. le.

Note that : we t tve etve ze the t ‘’ ute e the le eetvel.





9. DIRECTOR CIRCLE:

The lu the t teet tw eeul tet lle theDIRECTOR CIRCLE the

ve le. The et le le the et le hv u equl t 2 te the

l le.

10. EQUATION OF THE CHORD WITH A GIVEN MIDDLE POINT :

The equt the h the le ≡ + + + + = te t t

M

=

x g

y f

1

1

++

. Th lt e ut the

+

+ +

+ +

+ =

+

+

+

+

whh ete T = .

Note that : the htet h le thuh t ‘M’ e the le

e h whe le t M.

11. CHORD OF CONTACT :

I tw tet T & T

e w the t

t the le

≡ + + + + = the the equt the h tt TT

:

+ + + + + + = .

REMEMBER :

(a) h tt et l the t ‘’ t e .

(b) eth h tt TT

= 22 LR

R L2

+ .

(c) e the tle e the the tet & t h tt = 22

3

LR

LR

+

Whee the u the le & the leth the tet = .

(d) le etwee the tet

= t

− 22 R L

LR 2

whee = u ; = leth tet.

(e) Equt the le u the tle TT

:

+ +

+ = .

(f) The jt equt tet w the t

t the le

+ + + + = : = T.

Whee ≡ + + + + ; ≡

+

+

+

+

T ≡ + + + + + + .

12. POLE & POLAR :

(i) I thuh t the le the le thee e w tht le t eet the le

the lu the t teet the tet t & lle the POLAR

OF THE POINT P ; l lle the POLE OF THE POLAR .

(ii) The equt t the l t

w..t. the le + = ve

xx1

+ yy1 = a2 & the le eel the the equt the l ee

+

+ +

+ +

+ = . Nte tht the t

e the le the the

h tt tet & l wll e eeete the e equt.

(iii) le ve le + + = w..t. le + =

−−

CaB,

CaA

22

.





(iv) I the l t thuh t the the l e thuh .

(v) Tw le &

e jute eh the le

le

& ve ve ll tw t

& e t e jute eh the the l e thuh & ve-ve.

13. COMMON TANGENTS TO TWO CIRCLES :

(i) Whee the tw le ethe teet tuh eh the thee e FOU tet

tw the e tvee & the the e et tet.

(ii) Whe the teet thee e tw tet th the e et.(iii) Whe the tuh eh the :

(a) EXTERNALLY : thee e thee tet tw et e the tet t the

t tt .

(b) INTERNALLY : l e tet le t the t tt.

(iv) eth etel tet & tel tet t the tw le ve :

et

= 221

2 )r r (d −− & t

= 221

2 )r r (d +− .

Whee = te etwee the ete the tw le . &

e the the tw le.

(v) The et tet eet t t whh ve the le j ete le

etell the t the .Tvee tet eet t t whh ve the le j ete le

tell the t the .

14. RADICAL AXIS & RADICAL CENTRE :

The l tw le the lu t whe we w..t. the tw le e equl. The

equt l the tw le = &

= ve ;

= .e.

+

+

= .

NOTE THAT :

(a) I tw le teet the the l the h the tw le.

(b) I tw le tuh eh the the the l the tet the tw le t

the t tt.(c) l lw eeul t the le j the ete the tw le.

(d) l ee t lw thuh the t the le j the ete the tw

le.

(e) l et tet etwee the tw le.

(f) The t teet the l e thee le tke tw t te

lle the l ete thee le.

(g) te le eve tw whh hve the e l lle l te .

(h) le whh t hve l e et.

15. ORTHOGONALITY OF TWO CIRCLES :

Tw le = & = e t e thl t teet thll the tett the t teet lue ht le. The t tw le t e thl

:

+

=

+

.

Note :

(a) u the ete vle le thl t tw e le the l etwee the

tw e le .

(b) I tw le e thl the the l t '' t le w..t. the e le e

thuh the t whh the the e the ete thuh . Hee lu t whh

ve uh tht t l w..t. the le =

= &

= e uet le whh

thl t ll the thee le.





EXERCISE–I . Detee the tue the qultel e u le + – = ; – – = ; + + =

– + = . F the equt the le e u th qultel.

. le = w wth t ete t – t tuh the le + – + – =

etell. F the teet e the le = the te e.

. The le l +m +n = teet the uve + h + = t the t . The le

ete e thuh the . ve tht n + = l + m.

. Oe the ete the le u the etle D = + . I & e

the t – & eetvel the the e the etle.

. et e tht le thuh the

e the tht le + = . I the teet e

the le + + = &

e equl the the equt whh eeet

.

. le e thuh the t – . F the t the le the tet t

whh e llel t the tht le j t the ete.

. F the equt tht le whh thuh the teet the le =

+ = & ve the ueee the le + = t tw whe leth e

the t : .

.8 I the ve ue the le + = teet the - t

the t . The le = teet the - t the

t . t ve l the le = ve the -

teet the le t . F

The te the t the tle h the u e.

The te the t the le t .

The te the e the tle /th the e the tle .

. le w wth t ete the le + = t tuh the le – + = thuh

the t . F t equt.. t v u le + + + = wth ete ke w t ethe t the t

t the le ve l tet t the le thuh the t D – .

F the llw.

Equt the tet t .

te the t .

le D the u u te the t D the le.

v e qultel D the∆D.

v Equt the le u the ∆D l the teet e th le the

te e.

. F the lu the t the h le + = uh tht the eet teete the h the uve – – = ute ht le t the .

. F the equt le wth et uh tht the tw le + =

+ – – + = teet equl leth t.

. F the lu the le t t the tet t the le + = tete the

te e.

. Tet e w t the et le + = + = t ht le t e the.

hw tht the lu the t teet et le. F t u.

. F the equt t the le whh uh tht the leth the tet t t the t

e 7 2 eetvel.





. e le wth ete t the u . Fu le D eh wth uut ete – – eetvel e w. h the le tuhe the le e thuh the ete the le . I the leth th h e

eee x .

. I the vle le – + k = le etwee the le + – – + = + – – + = wthut teet tuh ethe le the the e k whee ∈ I. F the vlue – .

.8 Ot the equt the tht le thuh the t & k ° le wth thetet t t the le + + = . F the equt the le eh u

whe ete e thee tht le t te 2 .

. vle le e thuh the t & tuhe the -; hw tht the lu the the e the ete thuh = .

. F the lu the t ll h the le + = uh tht the le j & the t teet the h wth the le ke equl le wth .

. le wth ete the t qut tet t = + = – the -. et h k e

the ete the le. I the vlue h + k = + a b whee a u the vlue + .

. le tet t the the t qut t the t eetvel. D e llel tet t the le wth le – . I the t e the - whle

D e the - the e the ue D 2 q. ut the the u the le.

. le

e etell tet the e th tell tet t the le

. The

e eetvel the ete the thee le e lle. h

l tel tet

. Gve tht the leth the h

p

nm wheem n

p e tve tee m p e eltvel e n t vle the que e the vlue m + n + p.

. F the equt the le thuh the thee t 8. l thete the t teet the tet t the le t the t whee t ut thetht le + + = .

. The le – + = tet t le = t . I the u the le 13 . F the

equt the le .

. F the equt the le whh e thuh the t & whh tuhe the le + + = t the t t.

. F the equt the le whe u whh tuhe the le + – – – =tell t the t – – .

.8 Gve tht ht le tezu h e le. ve tht the leth the ht le le the H e the leth e.

. et K ete the que the ete the le whe ete the h thetw le + + + + = + + + + =

W ete the u the te t whee ll vle h theuve = 8 ute ht le t the e uet.

H ete the que the leth the tet the t the le + + – = .

F the vlue KWH.

. et =

= e tw le teet t th e tet t - le =

whee > . I ut the le = = 3

52

the the vlue .





EXERCISE–II

. hw tht the equt tht le eet the le + = tw t t equl te

'' t

t ueee

+

+ ( )2d

2 = .

. hu D h e leth . le wth ete '' e thuh the te vete

lkewe le wth ete e thuh D. I the tw le e tet t eh the the

e the hu.

. et e el ue uh tht

le ut le ete t .

t t e ee.

I the u vlue t – + t – + 2 b whee ∈ I the vlue

+ .

. ele ht le tle whe e e 2 le etel the t qut wth the

e the hteue the te e. I t le ve tht the lu t et

+ =

9

32.

. el ue te + = . I the u u vlue the eex7y4

z

−−=

e M eetvel the the vlue M + .

. The l the le + + + + = + + + 8 + = tuhe

the le ² + ² + + = . hw tht ethe = / = .

. F the equt the le thuh the t teet le + =

+ + + = & utt the le + = thll.

.8 The ete the le = le the le + = & = ut thll the le

+ = . hw tht le = e thuh tw e t & the te.

. F the equt le thuh the the le – + – = thlt t. I th le thl t the le + – k + k – 8= the the vlue k.

F the equt the le whh ut the le + – – 8 + = the te e thll.

. hw tht the lu the ete le whh ut tw ve le thll tht le

& hee eue the lu the ete the le whh ut the le + + + = &

+ + + = thll. Iteet the lu.

. F the equt le whh tuhe the le + = t the t ut the le

+ + + = thll.

. F the equt the le thuh the t – the we the t w..t. the

le t ut the le + – – – = thll.

. e l le thuh tw e t & . The the h

whh the le + – – – = ut the ee the l e uet t t.

F the te th t.

. F the equt le thuh el t the te l le tht e

tet t t the lu the t teet utull eeul tet t the le

+ = .

. The le : + + k + + k – k + = e thuh tw e t eve el ue

k. F

(i) the te thee tw t. (ii) the u vlue the u le .





. F the equt le whh -l wth le + + = &

+ + + + = . It ve tht the ete the le t e etee le the l

thee tw le.

. The le whh ut the l le thuh the e t ≡ ≡

thll thuh tw e t

whh e el . F

the vlue ( )3

2

3

1

3

2

3

1

yyxx +++ .

.8 F the equt le whh tuhe the le – 8 + = the le

+ – 8 – 8 = te the ve le.

. F the equt the le whh e thuh the eet the - thll & ut the

le + = t le º.

. e tw le u ''

u '' > th l the t qut

tuh the te e. I eh the t lte column-I the t a b ve

column-II.

Column-I Column-II

tuh eh the 22+

e thl

teet tht the h let 32 +

D e thuh the ete

223 +

T 223 −

EXERCISE–III

. The tle e the le + = . I hve -te &

eetvel the ∠ equl t

π/ π/ π/ D π/

I the le + + + k

+ = & + + k

+ k = teet thll

the ' k

' :

/ / / D /

[JEE ' ee +]

. Etete l etle e & . F the equt the tet t

the ule etle whh e llel t th l.

F the t the tht le = + whh eet t the le

+ + 8 = .

le u ut ll the utee the le + + =

tuh t

etell. F the lu the ete th ute le. l the equt the

tet the tw le whe the le j the ete the tw le le t

le º wth -. [EE ' M + + ]

. et e tet t the etete the ete le u . I

teet t t X the ueee the le the equl

PQ RS⋅ PQ RS+

2

2PQ RS

PQ RS

⋅+ D

( ) ( )PQ RS2 2

2

+

[ JEE ' ee ut ]

et + – = e the equt tet w the 'O' t le

u wth ete the t qut. I e the t tt the leth O.[JEE ' M ut ]





. F the equt the le whh e thuh the t teet le

+ – – + = + + – + = teet the le

+ + + + = thll. [EE ' M ut ]

Tet T T e w t T t the le + = . I the t T le the

le + q = the lu ete the ule tle T.

[ EE ' M ut ]

. I the tet t the t the le + + + = eet the tht le – + = t t the - the the leth

5 D 5

I > > the the tve vlue whh = – 2m1+ tet t

+ = – + =

22 b4a

b2

−

b2

b4a22 −

b2a

b2

−D

b2a

b

−[ JEE ' + ut ]

. The u the le hv ete t whe e the h ete the le

+ – – + = D 3 [JEE ' ]

. e + + = tet t le t -. Th le thl t le whh w

hv ete le eet wth e t – – . F equt le.

[JEE ' ut ]

.8 le ve + – = the le tuhe t etell l the - the the

lu t ete

{ : = } ∪ { : ≤ } { : + – = } ∪ { : ≤ }

{ : = } ∪ { : ≤ } D { : = } ∪ { : ≤ }

[JEE ' ]

. et D e qultel wth e 8 wth e llel t the e D = D. et D

e eeul t D. I le w e the qultel D tuh ll the e

the t u

/ D

Tet e w the t t the le + = .

tteet-: The tet e utull eeul.

because

tteet-: The lu the t whh utull eeul tet e w t the

ve le + = 8.

tteet- tue tteet- tue; tteet- et elt tteet-.

tteet- tue tteet- tue; tteet- NOT et elt tteet-.

tteet- tue tteet- le.

D tteet- le tteet- tue. [JEE +]

. e the tw uve

: = ;

: + – + = . The

tuh eh the l t e t

tuh eh the etl t tw t

teet ut t tuh t etl tw t

D

ethe teet tuh eh the





e : + + – = ;

: + + + =

whee el ue : + + – + = .

STATEMENT-1 : I le h le the le

t lw ete le .

and

STATEMENT-2 : I le ete le the le

t h le .

tteet- Tue tteet- Tue; tteet- et elt tteet-

tteet- Tue tteet- Tue; tteet- NOT et elt tteet- tteet- Tue tteet- Fle

D tteet- Fle tteet- Tue

Comprehension (3 questions together):

le u e equltel tle . The t tt wth the e

e D E F eetvel. The le ve the equt 3 + – = the

t D

2

3,

2

33. Futhe t ve tht the the ete e the e e the

le .

The equt le

– 32 + – = – 32 + +2

1 =

– 3 + + = D – 3 + – =

t E F e ve

2

3,

2

3 )0,3

2

1,

2

3 )0,3

23,

23

21,

23 D

23,

23

21,

23

Equt the e e

=3

2 + = –

3

2 – =

3

1 =

=2

3 + = –

2

3 – D = 3 =

[JEE 8 + + + + ]

. Tet w the t l 8 t the le + – – – =

tuh the le t the t . The equt the ule the tle

+ + – + = + – – + =

+ – + – = D + – – + =

The ete tw le

eh ut u e t te ut eh the. et

e the t the le eet j the ete

e le tuh le

etell. I tet t

thuh l tet t

the the u the le [JEE + ]





ANSWER SHEET

EXERCISE–I Q.1 que e ; + = ; + = Q.2 ze ze

Q.4 q. ut Q.5 = ; + = Q.6 & –

Q.7 = OR + =

Q.8 (i) (ii) 8 (iii)

Q.9 + – – + = OR + – + 8 – =

Q.10 – = ; + = ; – – ; ° )125 ± ut

v q. ut . q. ut; v + + + – teet ; teet

Q.11 + – – = Q.12 – – = Q.13 + =

Q.14 + = + ; = 22 ba + Q.15 + + – – = Q.16 Q.17

Q.18 = + = ; + = ; + + = ;

+ = ; + + + =

Q.20 + = Q.21 Q.22 = Q.23

Q.24 – + – – – = Q.25 + – + = OR + + – 8 + =

Q.26 + + + = Q.27 + – 8 – – = Q.29 Q.30 3

EXERCISE–II Q.2 q. ut Q.3 . Q.7 + + + – =

Q.8 ; – / / Q.9 + + – = ; k = ; + =

Q.10 + = ; l Q.11 + + + 8 = Q.12 + + – =

Q.13

3

23,2 Q.14 + + = Q.15 & //; =

22

1

Q.16 + + + – = Q.17 Q.18 + – – + =

Q.19

+

± 2

= Q.20 ; ; ; D

EXERCISE–III Q.1 (a) (b)

Q.2 (a) 8

+ = &

8

= ; (b) –/

(c) + + – = T: 0432yx3 =++− T

: 0432yx3 =−+− D..T.

T: 02y3x =−+ T

: 06y3x =++ T..T.

Q.3 (a) ; (b) O = + 10 Q.4 (a) + + – + = ; (b) + q =

Q.5 (a) ; (b) Q.6 Q.7 + – – + = Q.8 D Q.9 ;

Q.10 (a) ; (b)

; (c) D D Q.11 ; 8

Documents

Circles Key Concepts-1196