Write an equation for the ellipse.
-foci at $( - 2,5 ) , ( - 2 , - 1 )$; major axis length of 10
A) $\frac { ( x - 2 ) ^ { 2 } } { 25 } + \frac { ( y - 2 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( y - 2 ) ^ { 2 } } { 25 } + \frac { ( x + 2 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( y - 2 ) ^ { 2 } } { 25 } + \frac { ( x - 2 ) ^ { 2 } } { 16 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y - 2 ) ^ { 2 } } { 25 } = 1$

Question

Quizplus · Accepted Answer

The foci are at $(-2, 5)$ and $(-2, -1)$, which means the ellipse is vertically oriented (since the x-coordinates are the same and y-coordinates are different). The center of the ellipse is the midpoint of the foci, which is $(-2, (5 + (-1))/2) = (-2, 2)$. The major axis length is 10, so the semi-major axis length is $10/2 = 5$, which means $a^2 = 25$. Since the major axis is vertical, $a^2$ goes under the $y$ term. The distance between the center and either focus (the focal distance $c$) can be found by the distance formula or simply noting the difference in the y-values of the foci, which is $6/2 = 3$, so $c = 3$. To find $b^2$, we use the relationship $c^2 = a^2 - b^2$, giving $b^2 = 25 - 9 = 16$. Therefore, the correct equation is $\frac{(y - 2)^2}{25} + \frac{(x + 2)^2}{16} = 1$, which matches choice B.

Write an Equation for the Ellipse (−2,5),(−2,−1)( - 2,5 ) , ( - 2 , - 1 )(−2,5),(−2,−1)

Write an Equation for the Ellipse $( - 2,5 ) , ( - 2 , - 1 )$