10.1 Parabolas

10.1 Parabolas

A parabola is the set of all points (x,y) that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.

Vertex (h,k)

Focus (h, k + p)

Directrix y = k - p

Standard Equation of a Parabola

(x - h)2 = 4p(y - k) Vertical axis

Opens up (p is +) or down (p is -)

(y - k)2 = 4p(x - h) Horizontal axis

Opens right (p is +) or left (p is -)

p is the distance from the center to the focuspoint.

Ex. Find the vertex, focus, and directrix of the parabolaand sketch its graph. y2 + 4y + 8x - 12 = 0

y2 + 4y = -8x + 12 Now complete the square.

y2 + 4y + 4 = -8x + 12 + 4

(y + 2)2 = -8(x - 2)Write down the vertexand plot it. Then find p.

4p = -8

p = -2What does the negative pmean?

(y + 2)2 = -8x + 16

left

V(2,-2)F(0,-2)

Directrixx = 4

Ex. Find the standard form of the equation of theparabola with vertex (2,1) and focus (2,4).

First, plot the two points.

Which equation will we be using? Vert. or Horz. axis

Right, since the axis is vertical, we will be using

(x - h)2 = 4p(y - k)

What is p? p = 3

Now write down the equation.

(x - 2)2 = 12(y - 1)

Ellipses

Ellipses

Major axis Minor axis

Center point(h,k)

a

bc

a

a2 = b2 + c2

Focus point

FFVV

An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is constant.

Standard Equation of an Ellipse

1)()(

1)()(

2

2

2

2

2

2

2

2

=−

+−

=−

+−

aky

bhx

bky

ahx Horz. Major axis

Vert. Major axis

(h,k) is the center point.

The foci lie on the major axis, c units from the center.c is found by c2 = a2 - b2

Major axis has length 2a and minor axis has length 2b.

Sketch and find the Vertices, Foci, and Center point.Sketch and find the Vertices, Foci, and Center point.

x2 + 4y2 + 6x - 8y + 9 = 0

First, write the equation in standard form.

(x2 + 6x + ) + 4(y2 - 2y + ) = -9

(x2 + 6x + 9) + 4(y2 - 2y + 1) = -9 + 9 + 4

(x + 3)2 + 4(y - 1)2 = 4

11

)1(4

)3( 22

=−

++ yx

C (-3,1)

V (-1,1) (-5,1)

c2 = a2 - b2

c2 = 4 - 1

3=c

Foci are:

)1,33(&)1,33( +−−−

C (-3,1)

V (-1,1) (-5,1)

Eccentricity e of an ellipse measures the ovalness of the ellipse. e = c/a

In the last example, what is the eccentricity?

The smaller or closer to 0 that the eccentricity is, the more the ellipse looks like a circle.The closer to 1 the eccentricity is, the more elongated it is.

Find the center, vertices, and foci of the ellipse given by4x2 + y2 - 8x + 4y - 8=0

First, put this equation in standard form.

4(x2 - 2x + 1) + ( y2 + 4y + 4) = 8 + 4 + 4

4(x - 1)2 + (y + 2)2 = 16

( ) ( ) 116

241 22

=++− yx

C( , )a =b =c =

Vertices ( , ) ( , )Foci ( , ) ( , )e = Sketch it.

Hyperbolas

The standard form with center (h,k) is

( ) ( )

( ) ( ) 1

1

2

2

2

2

2

2

2

2

=−−−

=−−−

bhx

aky

bky

ahx

Note: a is under the positive term. It is not necessarilytrue that a is bigger than b.

Let’s take a look at the first hyperbola form.

C(h,k)

V(h+a,k)V(h-a,k)

a

b b

c is the distancefrom the center to the foci.

F(h+c,k)F(h-c,k)

Note: If c is the distance from the centerto F, and all radiiof a circle = , then the hyp. of the right triangle is also c.

Therefore, to find c,a2 + b2 = c2

c

Sketch the hyperbola whose equation is 4x2 - y2 = 16.

4x2 - y2 = 16 First divide by 16.

1164

22

=−yx

a = 2b = 4

Note: a is always underthe (+) term.

C(0,0)

Let’s sketch the hyperbola.

Write down a, b, c and the center pt.

Now find c.

52=c

FF VV

Now, we need to findthe equations of theasymptotes.

What are their slopesand one point that is on both lines?

224

±=±=m

)0(20 −±=− xy

( )0,52

)0,2(

±

±

F

V

Sketch the graph of 4x2 - 3y2 + 8x +16 = 0

4(x2 + 2x ) - 3y2 = -16+1 + 4

4(x + 1)2 - 3y2 = -12 Now, divide by -12 and switch the x and y terms.

( ) 131

4

22

=+− xy

C( , )a = b = c = e =Sketch

( )

27

7

3

20,1

=

=

=

=−

e

c

b

aC

V

V

F

V( , ) ( , )F( , ) ( , )

Eq. of asymptotes.

( )13

20 +±=− xyF

( )( )7,1

2,1

±−

±−

F

V

Classifying a conic from its general equation.

Ax2 + Cy2 + Dx + Ey + F = 0

If: A = C

AC = 0 , A = 0 or C = 0, but not both

AC > 0,

AC < 0

Both A and C = 0

CA≠

0≠