10.3 Ellipses

10.3 Ellipses Center point (h,k) Focus point V F a a c b F V Major axis a 2 = b 2 + c 2 Minor axis 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

(

x



h

) 2

a

2 (

x



h

) 2

b

2   (

y



a

2

k

) 2 (

y b

2 

k

) 2  1  1

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 c 2 = a 2 - b 2 Major axis has length 2a and minor axis has length 2b.

x 2 + 4y 2 + 6x - 8y + 9 = 0 First, write the equation in standard form.

(x 2 + 6x + ) + 4(y 2 - 2y + ) = -9 (x 2 + 6x + 9 ) + 4(y 2 - 2y + 1 ) = -9 + 9 + 4 (x + 3) 2 + 4(y - 1) 2 = 4

(

x

 3 ) 2 4  (

y

 1 ) 2 1  1

C (-3,1) V (-1,1) (-5,1)

c 2 = a 2 - b 2 c 2 = 4 - 1

c

 3

Foci are:

(  3  3 , 1 ) & (  3  3 , 1 )

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 by 4x 2 + y 2 - 8x + 4y - 8 = 0 First, put this equation in standard form.

4(x 2 - 2x + 1 ) + ( y 2 + 4y + 4 ) = 8 + 4 + 4 4(x - 1) 2 + (y + 2) 2 = 16



x

 4 1  

y

 16 2  2  1

C( , ) a = b = c = Vertices ( , ) ( , ) Foci ( , ) ( , ) e = Sketch it.

Assignment: 1-6 all, 7 - 29 odd