Chapter 10, Problem 49RE

To determine

Tofind:the new equation containing no x and y terms by rotating the axes, analyze and graph the new equation.

Answer to Problem 49RE

The equation which doesn’t contain x-y term is $\text{10}{\text{.0X}}^{\text{2}}\text{+5}{\text{.0Y}}^{\text{2}}\text{-20}\text{.0=0}$ .

Explanation of Solution

Given:

  ${\text{6x}}^{\text{2}}\text{+4xy}+9{\text{y}}^{\text{2}}\text{-20=0}$ .

Concept used:

The general equation of conic:

  ${\text{Ax}}^{\text{2}}{\text{+Bxy+Cy}}^{\text{2}}\text{+Dx+Ey+F=0}$

Rotation formula:

  $\text{x=Xcos}\left(\text{θ}\right)\text{-Ysin}\left(\text{θ}\right)$ .

  $\text{y=Xsin}\left(\text{θ}\right)\text{+Ycos}\left(\text{θ}\right)$ .

  $\text{cot}\left(\text{2}\left(\text{θ}\right)\right)\text{=}\frac{\text{A-C}}{\text{B}}$ .

Calculation:

The general equation of conic:

  ${\text{Ax}}^{\text{2}}{\text{+Bxy+Cy}}^{\text{2}}\text{+Dx+Ey+F=0}$ ………… $\left(\text{i}\right)$

According to the given:

  ${\text{6x}}^{\text{2}}\text{+4xy}+9{\text{y}}^{\text{2}}\text{-20=0}$ ………………. $\left(\text{ii}\right)$

Comparing $\left(\text{i}\right)$ and $\left(\text{ii}\right)$ :

  $\text{A=6,B=4 and C=9}$ .

  $\text{cot}\left(\text{2}\left(\text{θ}\right)\right)\text{=}\frac{\text{A-C}}{\text{B}}$ .

  $\text{θ=1}\text{.11}$ .

Rotation formula:

  $\text{x=Xcos}\left(\text{θ}\right)\text{-Ysin}\left(\text{θ}\right)$ .

  $\text{y=Xsin}\left(\text{θ}\right)\text{+Ycos}\left(\text{θ}\right)$ .

  $\text{x=Xcos}\left(1.11\right)\text{-Ysin}\left(1.11\right)$ .

  $\text{y=Xsin}\left(1.11\right)\text{+Ycos}\left(1.11\right)$ .

  $\text{x=0}\text{.447X-0}\text{.894Y}$

  $\text{y=0}\text{.894X+0}\text{.447Y}$ .

Upon substituting one another:

  $\text{10}{\text{.0X}}^{\text{2}}\text{+5}{\text{.0Y}}^{\text{2}}\text{-20}\text{.0=0}$ .

The graph of the above equation $\text{10}{\text{.0X}}^{\text{2}}\text{+5}{\text{.0Y}}^{\text{2}}\text{-20}\text{.0=0}$ will be:

  

The graph shows that at $\text{X=-1}\text{.414 and 1}\text{.414}$ ; $\text{Y=0}$ mean the equation has its zeroes at $\text{-1}\text{.414 and 1}\text{.414}$ .

Also, the graph shows that at $\text{Y=-2 and 2}$ ; $\text{X=0}$ .

Hence, the equation which doesn’t contain x-y term is $\text{10}{\text{.0X}}^{\text{2}}\text{+5}{\text{.0Y}}^{\text{2}}\text{-20}\text{.0=0}$ .