Question

The transformed equation of $4xy-3x^2=10$ when the axes are rotated through an angle $\theta$ such that tan$\theta$ = $2$ is .

• A. $X^2-4Y^2=10$
• B. $4X^2-Y^2=10$
• C. $XY-10=0$
• D. $2X^2-Y^2+10=0$

Answer 1

Answer: (A)

$X^2-4Y^2=10$

Given equation is $4xy-3x^2=10$
We know that
$x=X\cos \theta -Y\sin \theta$
$y=X\sin \theta +Y\cos \theta$

Given $\tan \theta =2$
$\Rightarrow \sin \theta =\frac{2}{\sqrt{5}},\cos \theta =\frac{1}{\sqrt{5}}$
$\Rightarrow x=\frac{1}{\sqrt{5}}(X-2Y)$ and $y=\frac{1}{\sqrt{5}}(2X+Y)$
So, the transformed equation is

$\frac{4}{5}(X-2Y)(2X+Y)-\frac{3}{5}(X-2Y{)}^2=10$
$\Rightarrow X^2-4Y^2=10$