Find an equation for the ellipse described.
-Vertices at $( - 4,5 )$ and $( 16,5 ) ;$ focus at $( 14,5 )$
A) $\frac { ( x + 6 ) ^ { 2 } } { 9 } + \frac { ( y + 2 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( x - 6 ) ^ { 2 } } { 49 } - \frac { ( y + 2 ) ^ { 2 } } { 19 } = 1$
C) $\frac { ( x - 6 ) ^ { 2 } } { 100 } + \frac { ( y - 5 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$

Question

Quizplus · Accepted Answer

The center of the ellipse is the midpoint of the vertices, which is $\left( \frac{-4 + 16}{2}, \frac{5 + 5}{2} \right) = (6, 5)$. The distance between the center and either vertex is the major radius $a$, which is $16 - 6 = 10$, so $a^2 = 100$. The distance between the center and a focus is $c$, which is $14 - 6 = 8$, so $c^2 = 64$. The minor radius $b$ can be found using the equation $b^2 = a^2 - c^2 = 100 - 64 = 36$. Therefore, the equation of the ellipse is $\frac{(x - 6)^2}{100} + \frac{(y - 5)^2}{36} = 1$.

Find an Equation for the Ellipse Described (−4,5)( - 4,5 )(−4,5)

Find an Equation for the Ellipse Described $( - 4,5 )$