Chapter 9.5, Problem 48HP

To determine

To Find: The Equation of hyperbola with focus at origin.

Answer to Problem 48HP

  $\frac{y^2}{36}-\frac{x^2}{16}=1$

Explanation of Solution

Given:

  $\text{Focus is at origin}\text{.}$

Formula Used:

  $\begin{array}{l}\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1\\ \frac{y^2}{a^2}-\frac{x^2}{b^2}=1\\ y=\pm \frac{b}{a}\left(x-h\right)+k\\ e=\frac{\sqrt{a^2+b^2}}{a}\\ l=\frac{2b^2}{a}\\ Vertices:\left(h-a,k\right)\&\left(h+a,k\right)\\ Foci:\left(h,k-c\right)\&\left(h,k+c\right)\end{array}$

  $\begin{array}{l}\text{Focus is at origin}\text{.}\\ \left(h,k\right)=\left(0,0\right)\\ Let,\\ a=6\Rightarrow a^2=36\\ b=4\Rightarrow b^2=16\\ \frac{y^2}{36}-\frac{x^2}{16}=1\end{array}$

Conclusion:

Thus, the equation of hyperbola is $\frac{y^2}{36}-\frac{x^2}{16}=1$ .