Question

Find the standard equation for the hyperbola with Vertices at (3,4) and (3,-2); Co-vertices at (7,1)and(-1,1). Please include steps!

Answer 1

Check the picture below. So based on those provided vertices the hyperbola looks pretty much like that one, with the center between the vertices of course

`\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=3\\ k=1\\ a=3\\ b=4 \end{cases}\implies \cfrac{(y-1)^2}{3^2}-\cfrac{(x-3)^2}{4^2}=1\implies \cfrac{(y-1)^2}{9}-\cfrac{(x-3)^2}{16}=1`