Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Center $( - 2,4 )$, focus $( - 2,11 )$, vertex $( - 2,6 )$
A) $\frac { ( y - 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 53 } = 1$
B) $\frac { ( y + 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$
C) $\frac { ( y + 4 ) ^ { 2 } } { 4 } + \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$
D) $\frac { ( y - 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$

Question

Quizplus · Accepted Answer

The center of the hyperbola is $(-2,4)$, indicating that the hyperbola is centered at this point, so the terms inside the parentheses should be $(y-4)$ and $(x+2)$. The vertex is $(-2,6)$, which is 2 units away from the center along the y-axis, indicating that $a=2$ and $a^2=4$. The focus is $(-2,11)$, 7 units away from the center, also along the y-axis, indicating that $c=7$ and $c^2=49$. Since $c^2=a^2+b^2$, we find $b^2=c^2-a^2=49-4=45$. Therefore, the correct equation is $\frac{(y-4)^2}{4} - \frac{(x+2)^2}{45} = 1$.

Find an Equation in Standard Form for the Hyperbola That (−2,4)( - 2,4 )(−2,4)

Find an Equation in Standard Form for the Hyperbola That $( - 2,4 )$