Conic sections

Claudio AlvaradoRylon Guidry

Erica Lux

Complete the Square

• Parabolas as well as other conic sections Parabolas as well as other conic sections are not always in the general form. The are not always in the general form. The general equation is Y=a(x-h)general equation is Y=a(x-h)2 2 + k.+ k.

• In order to get a conic into the general In order to get a conic into the general equation you must Complete the square equation you must Complete the square to change the equation of y = axto change the equation of y = ax22 +bx +c +bx +c into the general equation.into the general equation.

Look! A square!Look! A square!

Completing the Square

• Example: y=3x2- 18x –10

• Step 1:Isolate the x terms y=3x2- 18x –10 +10

+10y+10=3x2-18x

• Step 2: Divide by the x2 coefficient.

y+10=3x2-18x 3 3y+10=x2-6x 3

• Step 3: (a) divide the x coefficient by 2 then square it add the product to both sides of the equation

y+ 10= x2 +6x 3-6/2=(-3)2=9

y+10+9= x2-6x+9 3

Completing the squareCompleting the square

• Step 4: Factor the right hand side of the equation.y+10+9= x2-6x+9 3 y+10+9=(x-3)(x-3) 3 y+10+9=(x-3)2 3

• Step 5: Solve for y do that y=a(x-h)2 +k

y+10+9=(x-3)2 3

3{y+10}=3(x-3)2 3y+10+27=3(x-3)2

y+37= 3(x-3)2 -37 -37

y=3(x-3)2 -37

This is This is getting getting tough!!!tough!!!

ParabolasParabolas

ParabolaParabola-a set of all points in a plane that are -a set of all points in a plane that are the same distances from a given point called the same distances from a given point called the focus and a given line called the the focus and a given line called the directrixdirectrix

Latus RectumLatus Rectum- the line segment through focus - the line segment through focus and perpendicular to the axis of symmetryand perpendicular to the axis of symmetry

Parabola Graph

Directrix

Focus

Parabola

Form of Equation y=a(x-h)2 +k x=a(y-k)2+h

Axis of symmetry x=h y=k

Vertex (h,k) (h,k)

Focus (h,k+1/4a) (h+1/4a,k)

Directrix y=k-1/4a x=h-1/4a

Direction of opening

Upward if a>0

Down if a<0

Right if a>0

Left if a<0

Length of Latus Rectum

Abs (1/a) units Abs (1/a) units

CirclesCircles

Circle- the set of all points in a plane that Circle- the set of all points in a plane that are equal distances from a given point in are equal distances from a given point in the plane called the center.the plane called the center.

Radius-any segments whose endpoints are Radius-any segments whose endpoints are the center and a point on the circlethe center and a point on the circle

Equation of a circle:Equation of a circle:

(x-h)(x-h)22 + (y-k) + (y-k)22= r= r22

Center of a circle-(h,k)Center of a circle-(h,k)

Radius- rRadius- r

Pretty circle!Pretty circle!

CirclesFind the center and the radius

of a circle with and equation of x2+ y2+ 2x+ 4y-11=0

Step 1: Put all like terms together on the left hand side of the equation; place on constants on the right

x2+ y2+ 2x+ 4y-11=0

x2 + 2x + y2+ 4y =11

Step 2: Complete the Square x2 + 2x + y2+ 4y =11

x2+2x+1+y2+4y+4=11+1+4

Step 3: factorx2+2x+1+y2+4y+4=16(x+1)2+(y+2)2=16

Center = (-1,-2)Radius= 4

Finding Circle EquationsWrite an equation of a circle

whose endpoints of its diameter are at (-7,11) and (5,-10)

Step 1: Find the center by recalling the midpoint formula

(x1+x2, y1+y2)= (h,k) 2 2

(-7+5, 11-10)

2 2

Find the radius using the distance formula

D=((x2-x1)2+(y2-y1)2)1/2

D=((5-(-7))2+(-10-1)2)1/2

D=((12)2+(-21)2)1/2

D=(144+441)1/2

D=(585)1/2=24.187

Divide by 2 to find radius=12.093

Write the equation-

Center=(-1,.5) r2=146.41

(x+1)2+(y-k)2=146.41

Definition of an EllipseDefinition of an Ellipse

An ellipse is the set of all points in a plane such that the sum of the distances form the foci is constant.

4x2 + 9y2 + 16x -18y -11 = 0

EllipsesEllipses

Standard Equation for a center (0,0)

A) x2 + y2a2 + b2 =1

Major Axis is“x” because “a” under “x”

Foci (c,o) (-c,o) a2 >b2 b2 = a2 –c2

B) x2 + y2b2 + a2 =1

Major Axis is “y” because “a” under “y” foci (o,c) (o,-c)

True for both equations

Take me to Take me to your Ellipsesyour Ellipses

EllipsesFind the coordinates of

the foci and the length of the major and minor axis. Whose equations

is 16x2 + 4y2 = 144x2 + y2 or x2 + y2

a2 + b2 = 1 b2 + a2

16x2 + 4y2 = 144144 144 144

Since we know a2>b2 major axis is “y”

c=(27)1\2 c=(9)1\2

c=3(3)1\2

Length of your major axis= 2a =12

Length of your minor axis =2b =6

Foci (0,3(3)^1\2)(0,-3(3)^1\2)

b2 = a2 – c2 -27 = -c29 = 36 – c2 c2 = 27

Ellipses

When the center is Not at the origin (0,0) center(h,k)

Standard equation

A) (x-h)2

Ahh!!! Big Ahh!!! Big Big ellipse!!Big ellipse!!

HyperbolaHyperbola

DefinitionDefinition

• A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points, called the foci, is constant

HyperbolaHyperbolaStandard Equations of Standard Equations of

Hyperbolas with Center at Hyperbolas with Center at the Originthe Origin

• If a hyperbola has foci at (-c,o) and (c,o0 and if the absolute value of the difference of the distances from any point on the hyperbola to the tow foci is 2a units, then the standard equation of the hyperbola is x2 - y2

a2 - b2 =1, where c2 = a2+b2.

• If a hyperbola has foci at (o,-c) and (o,c) and if the absolute value of the difference of the distances from any point on the hyperbola to the two foci is 2a unit, and then the standard equation of the hyperbola is y2 - x2 a2 – b2 = 1, where c2= a2 + b2.

Ahhh!

Hyperbola

Equation of Hyperbola

x2 – y2

a2 b2=1

y2 – x2

a2 b2 =1

Equation of Asympote

b

Y=+/- ax

a

y = +/- bx

Transverse Axis horizontal vertical

Hyperbola

Standard Equations of Hyperbolas with Center at (h,k)

• The equation of a hyperbola with center at (h,k) and with a horizontal transverse axis x-h)2 - (y-k)2 a2 - b2 =1

• The equation of a hyperbola with center at (h,k) and with a vertical transverse axis is (y-k)2 - (x-h)2 a2 - b2 =1

ReferencesReferences

• Glencoe Algebra 2 textbook• Internet : www.glencoe.com

www.wwfhhh.com• Erica’s notes• Erica’s house

• Claudio’s house

Where’s Where’s Rylon’sRylon’s namename

And I And I did all did all this!this!

Roles

• Erica – Poster manager keeper dudette• Rylon – real life picture getter dude

• Claudio – with the help of Erica, did this wonderful presentation for you to behold

This marvelous project This marvelous project deserves a 100!!!!!deserves a 100!!!!!