Write an equation for the ellipse.
-foci at $( - 1,7 ) , ( - 1,1 )$; major axis length of 10
A) $\frac { ( y - 4 ) ^ { 2 } } { 25 } + \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( y - 4 ) ^ { 2 } } { 25 } + \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$

Question

Quizplus · Accepted Answer

The foci are vertically aligned (same x-coordinate, different y-coordinates), indicating the major axis is vertical. The center of the ellipse is the midpoint of the foci, which is $(-1, 4)$. The major axis length of 10 means the semi-major axis $a = 5$, so $a^2 = 25$. Since the major axis is vertical, $a^2$ goes under the $y$ term. The distance between the foci is 6, so $2c = 6$, giving $c = 3$. Using $c^2 = a^2 - b^2$, we find $b^2 = 16$. Thus, the correct equation is $\frac{(y-4)^2}{25} + \frac{(x+1)^2}{16} = 1$.

Write an Equation for the Ellipse (−1,7),(−1,1)( - 1,7 ) , ( - 1,1 )(−1,7),(−1,1)

Write an Equation for the Ellipse $( - 1,7 ) , ( - 1,1 )$