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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