Question

Find the vertices and foci of the hyperbola

Answer 1

`\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`

`\cfrac{y^2}{25}-\cfrac{x^2}{144}=1\implies \cfrac{(y-0)^2}{5^2}-\cfrac{(x-0)^2}{12^2}=1\hspace{5em} \begin{cases} h=0\\ k=0\\ a=5\\ b=12 \end{cases} \\\\\\ c=√(25+144)\implies c=13 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{rllll} vertices ~~ (0~~,~~\pm \stackrel{ a }{5})\\ foci ~~ (0~~,~~\pm \stackrel{c}{13}) \end{array}} \hfill ~`