Find the standard form of the equation of the ellipse with the given characteristics. foci: $( - 2,6 ) , ( - 2,10 ) \quad$ 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 major axis, which is $(-2,8)$. The length of the major axis is $17 - (-1) = 18$, so $a = 9$ and $a^2 = 81$. The distance between the center and either focus (c) is $10 - 8 = 2$, so $c^2 = 4$. Using $b^2 = a^2 - c^2$, we find $b^2 = 81 - 4 = 77$. The correct equation is $\frac{(x + 2)^2}{77} + \frac{(y - 8)^2}{81} = 1$.

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

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