Find the standard form of the equation of the hyperbola with the given characteristics. foci: asymptotes:

Question

Find the standard form of the equation of the hyperbola with the given characteristics. foci:  asymptotes:

Quizplus · Accepted Answer

The standard form of the equation of a hyperbola with foci (c,0) and (-c,0) and asymptotes y = mx and y = -mx is $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ where $a=c/m$ and $b=c\sqrt{m^2+1}$. Using the given foci, we have $c=10$ (since the distance between the foci is $2c$). Using the given asymptotes, we can rewrite them as $y=mx$ and $y=-mx$ (by rearranging), and then solve for $m$ using the fact that the slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$. We get $m=\pm \frac{4}{3}$. Choosing $m=\frac{4}{3}$ (we could also choose $m=-\frac{4}{3}$ and get the same answer), we have $a=\frac{5}{3}$ and $b=\frac{10\sqrt{10}}{3}$, so the standard form of the equation is $\boxed{\frac{(x-0)^2}{(\frac{5}{3})^2}-\frac{(y-0)^2}{(\frac{10\sqrt{10}}{3})^2}=1}$ or $\boxed{9x^2-100y^2=900}$.

Find the Standard Form of the Equation of the Hyperbola