Find the standard form of the equation of the ellipse with the given characteristics.
Foci: $( 7 , - 7 ) , ( 7 , - 1 )$ , endpoints of the major axis: $( 7 , - 13 ) , ( 7,5 )$
A) $\frac { ( x + 7 ) ^ { 2 } } { 72 } + \frac { ( y - 4 ) ^ { 2 } } { 81 } = 1$
B) $\frac { ( x - 7 ) ^ { 2 } } { 72 } + \frac { ( y + 4 ) ^ { 2 } } { 81 } = 1$
C) $\frac { ( x - 7 ) ^ { 2 } } { 81 } + \frac { ( y + 4 ) ^ { 2 } } { 72 } = 1$
D) $\frac { ( x - 7 ) ^ { 2 } } { 72 } + \frac { ( y + 4 ) ^ { 2 } } { 81 } = - 1$
E) $\frac { ( x - 4 ) ^ { 2 } } { 81 } + \frac { ( y + 7 ) ^ { 2 } } { 72 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics. 
Foci: $( 7 , - 7 ) , ( 7 , - 1 )$ , endpoints of the major axis: $( 7 , - 13 ) , ( 7,5 )$
A) $\frac { ( x + 7 ) ^ { 2 } } { 72 } + \frac { ( y - 4 ) ^ { 2 } } { 81 } = 1$
B) $\frac { ( x - 7 ) ^ { 2 } } { 72 } + \frac { ( y + 4 ) ^ { 2 } } { 81 } = 1$
C) $\frac { ( x - 7 ) ^ { 2 } } { 81 } + \frac { ( y + 4 ) ^ { 2 } } { 72 } = 1$
D) $\frac { ( x - 7 ) ^ { 2 } } { 72 } + \frac { ( y + 4 ) ^ { 2 } } { 81 } = - 1$
E) $\frac { ( x - 4 ) ^ { 2 } } { 81 } + \frac { ( y + 7 ) ^ { 2 } } { 72 } = 1$

Quizplus · Accepted Answer

The center of the ellipse is the midpoint of the major axis and the foci, which is $(7, -4)$. The length of the major axis is the distance between $(7, -13)$ and $(7, 5)$, which is 18 units, so $a=9$ and $a^2=81$. The distance between the foci is 6 units, so $c=3$ and $c^2=9$. Using $c^2=a^2-b^2$, we find $b^2=72$. Since the major axis is vertical, $a^2$ goes with $y$ and $b^2$ with $x$, leading to the equation $\frac{(x-7)^2}{72} + \frac{(y+4)^2}{81} = 1$.

Find the Standard Form of the Equation of the Ellipse (7,−7),(7,−1)( 7 , - 7 ) , ( 7 , - 1 )(7,−7),(7,−1)

Find the Standard Form of the Equation of the Ellipse $( 7 , - 7 ) , ( 7 , - 1 )$