Write the standard form of the equation of the hyperbola subject to the given conditions.
-Vertices: $( - 5 , - 1 ) , ( - 5 , - 5 )$; Foci $( - 5 , - 3 + \sqrt { 15 } ) , ( - 5 , - 3 - \sqrt { 15 } )$
A) $\frac { ( y - 3 ) ^ { 2 } } { 4 } - \frac { ( x - 5 ) ^ { 2 } } { 11 } = 1$
B) $\frac { ( y + 3 ) ^ { 2 } } { 4 } - \frac { ( x + 5 ) ^ { 2 } } { 11 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 4 } - \frac { ( y + 5 ) ^ { 2 } } { 11 } = 1$
D) $\frac { ( x + 5 ) ^ { 2 } } { 11 } - \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$

Question

Quizplus · Accepted Answer

The vertices and foci are aligned vertically, indicating the major axis is vertical. The center of the hyperbola is midway between the vertices and foci, at $(-5, -3)$. The distance between the center and either vertex (the semi-major axis length, $a$) is 2, so $a^2 = 4$. The distance from the center to either focus (the focal length, $c$) is $\sqrt{15}$, so $c^2 = 15$. The semi-minor axis length $b$ can be found using $c^2 = a^2 + b^2$, giving $b^2 = 11$. The correct equation, considering the vertical major axis, is $\frac{(y + 3)^2}{4} - \frac{(x + 5)^2}{11} = 1$.

Write the Standard Form of the Equation of the Hyperbola (−5,−1),(−5,−5)( - 5 , - 1 ) , ( - 5 , - 5 )(−5,−1),(−5,−5)

Write the Standard Form of the Equation of the Hyperbola $( - 5 , - 1 ) , ( - 5 , - 5 )$