Question

The equation $4xy-3x^2=a^2$ become when the axes are turned through an angle ${\tan }^{-1}2$ is

• A. $x^2+4y^2=a^2$
• B. $x^2-4y^2=a^2$
• C. $4x^2+y^2=a^2$
• D. $4x^2-y^2=a^2$

Answer 1

The correct option is B $x^2-4y^2=a^2$

Let $(X,Y)$ be the new coordinates of $(x,y)$.

Angle of rotation, $\theta ={\tan }^{-1}2$

$\Rightarrow \tan \theta =2\Rightarrow \sin \theta =\frac{2}{\sqrt{5}},\cos \theta =\frac{1}{\sqrt{5}}$

Now,

$x=X\cos \theta -Y\sin \theta =\frac{X-2Y}{\sqrt{5}}$

$y=X\sin \theta +Y\cos \theta =\frac{2X+Y}{\sqrt{5}}$

Therefore, transformed equation will be-

$4\left(\frac{X-2Y}{\sqrt{5}}\right)\left(\frac{2X+Y}{\sqrt{5}}\right)-3{\left(\frac{X-2Y}{\sqrt{5}}\right)}^2=a^2$

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

$\Rightarrow X^2-4Y^2=a^2$