Chapter 10, Problem 48RE

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 48RE

The equation which doesn’t contain x-y term is $\text{4}{\text{.5Y}}^{\text{2}}\text{-0}{\text{.5X}}^{\text{2}}\text{-4}\text{.5=0}$ .

Explanation of Solution

Given:

  ${\text{2x}}^{\text{2}}\text{-5xy}+{\text{2y}}^{\text{2}}\text{-}\frac{\text{9}}{\text{2}}\text{=0}$ .

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:

  ${\text{2x}}^{\text{2}}\text{-5xy}+{\text{2y}}^{\text{2}}\text{-}\frac{\text{9}}{\text{2}}\text{=0}$ ………………. $\left(\text{ii}\right)$

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

  $\text{A=2,B=-5 and C=2}$ .

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

  $\text{θ=0}\text{.785}$ .

Rotation formula:

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

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

  $\text{x=Xcos}\left(0.785\right)\text{-Ysin}\left(0.785\right)$ .

  $\text{y=Xsin}\left(0.785\right)\text{+Ycos}\left(0.785\right)$ .

  $\text{x=0}\text{.707X-0}\text{.707Y}$

  $\text{y=0}\text{.707X+0}\text{.707Y}$ .

Upon substituting one another:

  $\text{4}{\text{.5Y}}^{\text{2}}\text{-0}{\text{.5X}}^{\text{2}}\text{-4}\text{.5=0}$ .

The graph of the above equation $\text{4}{\text{.5Y}}^{\text{2}}\text{-0}{\text{.5X}}^{\text{2}}\text{-4}\text{.5=0}$ will be:

  

The graph shows that at $\text{x=0}$ $\text{y=1,-1}$ mean the equation has no zeroes.

Hence, the equation which doesn’t contain x-y term is $\text{4}{\text{.5Y}}^{\text{2}}\text{-0}{\text{.5X}}^{\text{2}}\text{-4}\text{.5=0}$ .