Question

A hyperbola centered at the origin has a vertex at (9, 0) and a focus at (−15, 0). Which are the equations of the asymptotes? y = ±x y = ±x y = ±x y = ±x

Answer 1

Answer:

The correct answer in C. y=+-4/3x

Step-by-step explanation:

Answer 2

Well, we know is centered at the origing, thus h,k are just 0,0.

using the provided vertex and focus point, gives us a distance for "a" of 9 and a distance for "c" of 15, check the picture below.

$\bf \textit{hyperbolas, horizontal traverse axis }\\\\ \cfrac{(x- h)^2}{ a^2}-\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2+ b ^2}\\ \textit{asymptotes}\quad y= k\pm \cfrac{b}{a}(x- h) \end{cases}\\\\ -------------------------------$

$\bf \begin{cases} a=9\\ c=15 \end{cases}\implies c=\sqrt{a^2+b^2}\implies \sqrt{c^2-a^2}=b \\\\\\ \sqrt{15^2-9^2}=b\implies \sqrt{144}=b\implies \boxed{12=b}\\\\ -------------------------------\\\\ asymptotes\qquad y=0\pm\cfrac{12}{9}(x-0)\implies y=\pm\cfrac{4}{3}x$