Question

The center of a hyperbola is (−2,4) , and one vertex is (−2,7) . The slope of one of the asymptotes is 1/2 .

Answer 1

Answer:

The equation is $\frac{(y - 4)^{2} }{9 } - \frac{(x + 2)^{2} }{36 } = 1$.

Step-by-step explanation:

The equation of a hyperbola is represented by $\frac{(y - k)^{2} }{b^{2} } - \frac{(x - h)^{2} }{a^{2} } = 1$, where (h, k) is the center of the hyperbola.

As per the given condition, h = -2 and k = 4.

Thus, the equation becomes $\frac{(y - 4)^{2} }{b^{2} } - \frac{(x + 2)^{2} }{a^{2} } = 1$.

One vertex is (-2, 7).

Hence, putting x = -2 and y = 7, we get $b^{2} = 9$.

The slope of the asymptote is $\frac{b}{a}$.

Hence, $\frac{b^{2} }{a^{2} } = \frac{1}{4} \\a^{2} = 36$.

Thus, the equation is $\frac{(y - 4)^{2} }{9 } - \frac{(x + 2)^{2} }{36 } = 1$.