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 major axis and the foci, which is at $(-5, -8)$. The major axis length is the distance between the endpoints of the major axis, which is 16 units (from $-16$ to $0$ on the y-axis), so the semi-major axis length $a = 8$, and $a^2 = 64$. The distance between the foci is 8 units (from $-12$ to $-4$ on the y-axis), so the focal distance $2c = 8$, and $c = 4$, $c^2 = 16$. Using $c^2 = a^2 - b^2$, we find $b^2 = a^2 - c^2 = 64 - 16 = 48$. Since the major axis is vertical, $a^2$ goes with the $y$ term and $b^2$ with the $x$ term, leading to the equation $\frac{(x + 5)^2}{48} + \frac{(y + 8)^2}{64} = 1$.

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 $( - 5 , - 12 ) , ( - 5 , - 4 )$