ellipses

1. (x – h)2
(y – k)2
a2
b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
(3, -1) (7, -1)(-1, -1)
(3, -3)
(3, 1)
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
42
22+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
So the right point is (7, –1), the top
point is (3, 1), the left and bottom
points are (–1, –1) and (3, –3).
The Standard Form
(of Ellipses)

2. 9(x – 1)2
4(y – 2)2
36 4 36 9
Example B. Put 9x2
+ 4y2
– 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2
– 18x + 4y2
– 16y = 11 factor out the square-coefficients
9(x2
– 2x ) + 4(y2
– 4y ) = 11 complete the square
9(x2
– 2x + 1 ) + 4(y2
– 4y + 4 ) = 11 + 9 + 16
+9 +16
+ = 1
(x – 1)2
(y – 2)2
22
32
+ = 1
Ellipses
9(x – 1)2
+ 4(y – 2)2
= 36 divide by 36 to get 1
Hence, Center: (1, 2),
x-radius is 2,
y-radius is 3.
(-1, 2) (3, 2)
(1, 5)
(1, -1)
(1, 2)