Question

1. Find the vertices and locate the foci for the hyperbola whose equation is given.

Answer 1

$\bf \cfrac{(x-{{ h}})^2}{{{ a}}^2}-\cfrac{(y-{{ k}})^2}{{{ b}}^2}=1 \qquad center\ ({{ h}},{{ k}})\qquad vertices\ ({{ h}}\pm a, {{ k}})\\\\ -----------------------------\\\\ \textit{now let's take a look at yours} \\\\\\ 49x2 - 16y2 = 784\implies \cfrac{49x^2}{784}-\cfrac{16y^2}{784}=1 \\\\\\ \cfrac{x^2}{16}-\cfrac{y^2}{49}=1\implies \cfrac{(x-0)^2}{4^2}-\cfrac{(y-0)^2}{7^2}=1 \\\\\\ recall\implies center\ ({{ h}},{{ k}})\qquad vertices\ ({{ h}}\pm a, {{ k}})$

$\bf \textit{now, for the foci, the foci are "c" distance from the center point}\\\\\ whereas\qquad c=\sqrt{a^2+b^2}\qquad \textit{ that is }\qquad h\pm \sqrt{a^2+b^2}$

notice your "a" and "b" components, to get the distance "c" from the center to either foci and the vertices, of course, are h + a, k and h - a, k