Find an equation in standard form for the ellipse that satisfies the given conditions.
-An ellipse with foci at $( - 1,3 )$ and $( 5,3 )$; major axis length of 10
A) $\frac { ( x + 3 ) ^ { 2 } } { 25 } + \frac { ( x + 2 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( x - 2 ) ^ { 2 } } { 25 } + \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( y - 3 ) ^ { 2 } } { 25 } + \frac { ( x - 2 ) ^ { 2 } } { 16 } = 1$
D) $\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 16 } = 1$

Question

Quizplus · Accepted Answer

The center of the ellipse is the midpoint of the foci, which is $(\frac{-1+5}{2}, \frac{3+3}{2}) = (2, 3)$. The major axis length is 10, so $2a = 10$ which means $a = 5$ and $a^2 = 25$. Since the foci and the major axis are horizontal, $a^2$ goes under the $x$ term and the minor axis $b^2$ will be under the $y$ term. The correct equation is $\frac{(x-2)^2}{25} + \frac{(y-3)^2}{16} = 1$, where $b^2 = 16$ is derived from the fact that the distance between the center and the foci (c) is 3 (half the distance between the foci), and $a^2 = b^2 + c^2$.

Find an Equation in Standard Form for the Ellipse That (−1,3)( - 1,3 )(−1,3)

Find an Equation in Standard Form for the Ellipse That $( - 1,3 )$