Question

Write an equation of a hyperbola with the given foci and vertices.

foci (0,+/-25) ; vertices (0,+/-7)

Answer 1

Check the picture below.

so the hyperbola is more or less like that, with a center at the origin, we know "c" and we also know "a", let's find "b".

`\textit{hyperbolas, vertical traverse axis } \\\\ \cfrac{(y- k)^2}{ a^2}-\cfrac{(x- h)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad √( a ^2 + b ^2) \end{cases} \\\\[-0.35em] ~\dotfill`

`\begin{cases} h=0\\ k=0\\ a=7\\ c=25 \end{cases}\implies \cfrac{(y- 0)^2}{ 7^2}-\cfrac{(x- 0)^2}{ b^2}=1 \\\\\\ c=√(a^2+b^2)\implies c^2=a^2+b^2\implies c^2-a^2=b^2\implies 25^2 - 7^2 = b^2 \\\\\\ 576=b^2\hspace{9em}\cfrac{(y- 0)^2}{ 7^2}-\cfrac{(x- 0)^2}{ 576}=1\implies \boxed{\cfrac{y^2}{49}-\cfrac{x^2}{576}=1}`