Chapter 8.5, Problem 38PS

Solution

To find: An equation for the cross section of the mirror if the transverse axis is horizontal.

The equation for the cross section of the mirror if the transverse is horizontal is $\frac{x^2}{1089}-\frac{y^2}{2047}=1$ .

Given information :

A satellite is carrying a telescope that has a hyperbolic mirror for which $a=36\text{ cm and }c=56\text{ cm}$ .

Formula used:

The distance between two points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ is given by: $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Explanation :

To write the equation, find the value of $b^2$ .

  $\begin{array}{c}{b}^2=c^2-a^2\\ ={\left(56\right)}^2-{\left(33\right)}^2\\ =3136-1089\\ =2047\end{array}$

Substitute $a=33\text{ and }{b}^2=2047$ into the equation of hyperbola with horizontal transverse axis.

  $\begin{array}{c}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \frac{x^2}{{\left(33\right)}^2}-\frac{y^2}{2047}=1\\ \frac{x^2}{1089}-\frac{y^2}{2047}=1\end{array}$

Therefore, the equation for the cross section of the mirror if the transverse is horizontal is $\frac{x^2}{1089}-\frac{y^2}{2047}=1$ .