For ellipses, a >= ba≥b (when a = ba=b, we have a circle)
aa represents half the length of the major axis while bb represents half the length of the minor axis.
This means that the endpoints of the ellipse's major axis are aa units (horizontally or vertically) from the center (h, k)(h,k) while the endpoints of the ellipse's minor axis are bb units (vertically or horizontally)) from the center.
The ellipse's foci can also be obtained from aa and bb.
An ellipse's foci are ff units (along the major axis) from the ellipse's center
where f^2 = a^2 - b^2f2=a2−b2
Example 1:
x^2/9 + y^2/25 = 1x29+y225=1
a = 5a=5
b = 3b=3
(h, k) = (0, 0)(h,k)=(0,0)
Since aa is under yy, the major axis is vertical.
So the endpoints of the major axis are (0, 5)(0,5) and (0, -5)(0,−5)
while the endpoints of the minor axis are (3, 0)(3,0) and (-3, 0)(−3,0)
the distance of the ellipse's foci from the center is
f^2 = a^2 - b^2f2=a2−b2
=> f^2 = 25 - 9⇒f2=25−9
=> f^2 = 16⇒f2=16
=> f = 4⇒f=4
Therefore, the ellipse's foci are at (0, 4)(0,4) and (0, -4)(0,−4)
Example 2:
x^2/289 + y^2/225 = 1x2289+y2225=1
x^2/17^2 + y^2/15^2 = 1x2172+y2152=1
=> a = 17, b = 15⇒a=17,b=15
The center (h, k)(h,k) is still at (0, 0).
Since aa is under xx this time, the major axis is horizontal.
The endpoints of the ellipse's major axis are at (17, 0)(17,0) and (-17, 0)(−17,0).
The endpoints of the ellipse's minor axis are at (0, 15)(0,15) and (0, -15)(0,−15)
The distance of any focus from the center is
f^2 = a^2 - b^2f2=a2−b2
=> f^2 = 289 - 225⇒f2=289−225
=> f^2 = 64⇒f2=64
=> f = 8⇒f=8
Hence, the ellipse's foci are at (8, 0)(8,0) and (-8, 0)(−8,0)
