Chapter 9, Problem 36CR

To determine

To find: The standard equation of hyperbola

Answer to Problem 36CR

The standard equation of the hyperbola is $\frac{y^2}{1}-\frac{x^2}{3}=1$

Explanation of Solution

Given information:

The vertices of hyperbola is $\left(0,\pm 1\right)$

The foci of hyperbola is $\left(0,\pm 2\right)$

Formula used:

The general equation of the hyperbola for the vertical transverse axis when center is at $\left(0,0\right)$ is $\frac{y^2}{a^2}-\frac{x^2}{b^2}$

  $c^2=a^2+b^2$

Calculation:

The center of the hyperbola is at the origin. The distance between one of the foci and the center is 3 so $c=3$ .

The vertices lie on the vertical line, so the hyperbola has a vertical transverse axis. The distance between one of the vertices and the center is 1 so $a=1$

Furthermore, $c=3$ and $a=1$ , and it follows that

  $\begin{array}{l}b=\sqrt{c^2-a^2}\\ b=\sqrt{2^2-1^2}\\ b=\sqrt{4-1}\\ b=\sqrt{3}\end{array}$

So, the hyperbola is vertical transverse axis and the standard form of the equation of the hyperbola is

  $\frac{y^2}{1^2}-\frac{x^2}{{\left(\sqrt{3}\right)}^2}=1$

This equation simplifies to

  $\frac{y^2}{1}-\frac{x^2}{3}=1$

Therefore, the standard equation of the hyperbola is

  $\frac{y^2}{1}-\frac{x^2}{3}=1$