Question

Find the equation of the hyperbola with the following properties. Express your answer in standard form.Foci at (7, 0) and (7, 10)Asymptotes ofy -5 = t;( -7

Answer 1

Answer:

$\frac{(y-5)^2}{16}-\frac{(x-7)^2}{9}=1$

Explanation:

The standard form of an hyperbola is:

$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$

Where (h, k) are the coordinates of the center.

We are given the asyptotes and the foci.

The foci are (7, 0) and (7, 10)

The y value of the center of the parabola is midway from the two foci. Then, the y-coordinate of the center is 5

The coordinated of the center are (7, 5)

Now, we can use that the form of the asymptotes are:

$y=k\pm\frac{a}{b}(x-h)$

We have:

$y-5=\frac{4}{3}(x-7)$

Then:

$\frac{a}{b}=\frac{4}{3}$

a = 4

b = 3

Now we can write:

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