Chapter 9.3, Problem 83E

To determine

The angle through which the angle is rotated and use a graphing utility to graph the conic.

Answer to Problem 83E

The angle through which the angle is rotated is $\theta ={28.15}^{\circ }$ .

Explanation of Solution

Given information:

The given equation is $24x^2+18xy+12y^2=34$ .

Calculation:

The general second-degree equation

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

where $B\ne 0$ , can be written as

  $A\text{'}{\left(x\text{'}\right)}^2+C\text{'}{\left(y\text{'}\right)}^2+D\text{'}x\text{'}+E\text{'}y\text{'}+F\text{'}=0$

by rotating the coordinate axes through an angle $\theta$ , where

  $\cot 2\theta =\frac{A-C}{B}$

Here, $A=24$ , $B=18$ and $C=12$ . Therefore,

  $\begin{array}{l}\cot 2\theta =\frac{24-12}{18}\\ \cot 2\theta =\frac{12}{18}\\ \cot 2\theta =\frac{2}{3}\\ \tan 2\theta =\frac{3}{2}\\ 2\theta ={\tan }^{-1}\frac{3}{2}\\ \theta =\frac{{\tan }^{-1}\frac{3}{2}}{2}\\ \theta ={28.15}^{\circ }\end{array}$

The graph of the equation is drawn below:

  

Hence,

The angle through which the angle is rotated is $\theta ={28.15}^{\circ }$ .