Write an equation for the hyperbola.
-vertices at $( - 4,3 )$ and $( 2,3 )$, passing through the point $( - 6,9 )$
A) $\frac { ( x + 1 ) ^ { 2 } } { 9 } - \frac { ( y - 3 ) ^ { 2 } } { \frac { 117 } { 4 } } = 1$
B) $\frac { ( x + 1 ) ^ { 2 } } { 9 } - \frac { ( y - 3 ) ^ { 2 } } { \frac { 81 } { 4 } } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { \frac { 117 } { 4 } } - \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$
D) $\frac { ( x + 1 ) ^ { 2 } } { \frac { 81 } { 4 } } - \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$

Question

Quizplus · Accepted Answer

The vertices indicate that the hyperbola is centered at the midpoint of the vertices, which is $(-4 + 2)/2, (3 + 3)/2 = (-1, 3)$. The distance between the center and either vertex is the length of the semi-major axis, $a$, which is $2 - (-1) = 3$, so $a^2 = 9$. Since the vertices are horizontal from each other, the $x$ term is positive in the equation, and the $x$-axis term comes first. To find $b^2$, we use the point $(-6, 9)$ that the hyperbola passes through. Substituting the center and the point into the hyperbola equation $\frac{(x + 1)^2}{a^2} - \frac{(y - 3)^2}{b^2} = 1$, we solve for $b^2$, which gives $b^2 = \frac{81}{4}$. Therefore, the correct equation is $\frac{(x + 1)^2}{9} - \frac{(y - 3)^2}{\frac{81}{4}} = 1$.

Write an Equation for the Hyperbola (−4,3)( - 4,3 )(−4,3) And (2,3)( 2,3 )(2,3)

Write an Equation for the Hyperbola $( - 4,3 )$ And $( 2,3 )$