Find the standard form of the equation of the ellipse with the given characteristics.
vertices: $( - 6 , - 8 ) , ( - 6,6 ) \quad$ minor axis of length: 10
A) $\frac { ( x - 6 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 49 } = 1$
B) $\frac { ( x + 6 ) ^ { 2 } } { 25 } + \frac { ( y + 1 ) ^ { 2 } } { 49 } = 1$
C) $\frac { ( x + 6 ) ^ { 2 } } { 49 } + \frac { ( y + 1 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( x + 1 ) ^ { 2 } } { 49 } + \frac { ( y + 6 ) ^ { 2 } } { 25 } = 1$
E) $\frac { ( x - 1 ) ^ { 2 } } { 49 } + \frac { ( y - 6 ) ^ { 2 } } { 25 } = 1$

Question

Quizplus · Accepted Answer

The vertices given are $(-6, -8)$ and $(-6, 6)$, which are aligned vertically, indicating that the major axis is vertical. The distance between the vertices is $6 - (-8) = 14$, so the length of the major axis is 14, and thus the semi-major axis $a = 14/2 = 7$, giving $a^2 = 49$. The minor axis is given as 10, so the semi-minor axis $b = 10/2 = 5$, giving $b^2 = 25$. The center of the ellipse is the midpoint of the vertices, which is $(-6, (6 + (-8))/2) = (-6, -1)$. Therefore, the standard form of the equation is $\frac{(x + 6)^2}{25} + \frac{(y + 1)^2}{49} = 1$, matching option B.

Find the Standard Form of the Equation of the Ellipse (−6,−8),(−6,6)( - 6 , - 8 ) , ( - 6,6 ) \quad(−6,−8),(−6,6)

Find the Standard Form of the Equation of the Ellipse $( - 6 , - 8 ) , ( - 6,6 ) \quad$