Find the standard form of the equation of the ellipse with the given characteristics.
Foci: $( - 5 , - 12 ) , ( - 5 , - 4 )$ , endpoints of the major axis: $( - 5 , - 16 ) , ( - 5,0 )$
A) $\frac { ( x + 5 ) ^ { 2 } } { 64 } + \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$
B) $\frac { ( x - 5 ) ^ { 2 } } { 48 } + \frac { ( y - 8 ) ^ { 2 } } { 64 } = 1$
C) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 5 ) ^ { 2 } } { 48 } = 1$
D) $\frac { ( x + 8 ) ^ { 2 } } { 64 } + \frac { ( y + 5 ) ^ { 2 } } { 48 } = 1$
E) $\frac { ( x + 5 ) ^ { 2 } } { 48 } + \frac { ( y + 8 ) ^ { 2 } } { 64 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics. 
Foci: $( - 5 , - 12 ) , ( - 5 , - 4 )$ , endpoints of the major axis: $( - 5 , - 16 ) , ( - 5,0 )$ 
A) $\frac { ( x + 5 ) ^ { 2 } } { 64 } + \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$
B) $\frac { ( x - 5 ) ^ { 2 } } { 48 } + \frac { ( y - 8 ) ^ { 2 } } { 64 } = 1$
C) $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 5 ) ^ { 2 } } { 48 } = 1$
D) $\frac { ( x + 8 ) ^ { 2 } } { 64 } + \frac { ( y + 5 ) ^ { 2 } } { 48 } = 1$
E) $\frac { ( x + 5 ) ^ { 2 } } { 48 } + \frac { ( y + 8 ) ^ { 2 } } { 64 } = 1$

Quizplus · Accepted Answer

The center of the ellipse is the midpoint of the foci and the major axis, which is $(-5, -8)$. The major axis length is 16 (from -16 to 0 on the y-axis), so $a=8$ and $a^2=64$. The distance between the foci is 8 (from -12 to -4 on the y-axis), so $c=4$ and $c^2=16$. Using $b^2=a^2-c^2$, we find $b^2=48$. Since the major axis is vertical, $a^2$ goes with the $y$ term, resulting in option E.

Find the Standard Form of the Equation of the Ellipse (−5,−12),(−5,−4)( - 5 , - 12 ) , ( - 5 , - 4 )(−5,−12),(−5,−4)

Find the Standard Form of the Equation of the Ellipse $( - 5 , - 12 ) , ( - 5 , - 4 )$