Find the equation of the hyperbola with center $( 2 , - 2 ) , a ^ { 2 } = 25$ and $b ^ { 2 } = 4$.
A) $\quad \frac { ( x - 2 ) ^ { 2 } } { 25 } - \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$
B) $\quad \frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$
C) $\quad \frac { ( x - 2 ) ^ { 2 } } { 25 } - \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$
D) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$
E) $\quad \frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$

Question

Quizplus · Accepted Answer

The general equation for a hyperbola with center $(h,k)$, horizontal axis length $2a$, and vertical axis length $2b$ is:
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
Substituting in the given values, we get:
$$\frac{(x-2)^2}{5^2}-\frac{(y+2)^2}{2^2}=1$$
Simplifying, we get:
$$\frac{(x-2)^2}{25}-\frac{(y+2)^2}{4}=1$$
So, the correct choice is (C).

Find the Equation of the Hyperbola with Center (2,−2),a2=25( 2 , - 2 ) , a ^ { 2 } = 25(2,−2),a2=25

Find the Equation of the Hyperbola with Center $( 2 , - 2 ) , a ^ { 2 } = 25$