Find the standard form of the equation of the ellipse with the given characteristics.
foci: $( - 2,6 ) , ( - 2,10 )$
endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$
A) $\frac { ( x - 2 ) ^ { 2 } } { 77 } + \frac { ( y + 8 ) ^ { 2 } } { 81 } = 1$
B) $\frac { ( x + 2 ) ^ { 2 } } { 77 } + \frac { ( y - 8 ) ^ { 2 } } { 81 } = 1$
C) $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y - 8 ) ^ { 2 } } { 77 } = 1$
D) $\frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 77 } = 1$
E) $\frac { ( x + 8 ) ^ { 2 } } { 81 } + \frac { ( y - 2 ) ^ { 2 } } { 77 } = 1$

Question

Quizplus · Accepted Answer

The center of the ellipse is the midpoint of the foci and the major axis, which is $(-2,8)$. The length of the major axis is the distance between $(-2,-1)$ and $(-2,17)$, which is $18$, so $a=9$ and $a^2=81$. The distance between the foci is $4$, so $c=2$ and $c^2=4$. Using $c^2=a^2-b^2$, we find $b^2=a^2-c^2=81-4=77$. The standard form of the ellipse equation is $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$, where $(h,k)$ is the center of the ellipse. Substituting the values, we get $\frac{(x+2)^2}{77}+\frac{(y-8)^2}{81}=1$.

Find the Standard Form of the Equation of the Ellipse (−2,6),(−2,10)( - 2,6 ) , ( - 2,10 )(−2,6),(−2,10)

Find the Standard Form of the Equation of the Ellipse $( - 2,6 ) , ( - 2,10 )$