Question

If the axes be turned through an angle $\tan ^{-1} 2 .$ What does the equation $4 \,x y-3 x^{2}=a^{2}$ become?

• A. $X^{2}-4 Y^{2}=a^{2}$
• B. $X^{2}+4 Y^{2}=a^{2}$
• C. $X^{2}+4 Y^{2}=-a^{2}$
• D. None of these

Answer 1

Answer: (A)

Here, $\tan\, \theta=2$,
So, $\cos \theta=\frac{1}{\sqrt{5}}, \sin \theta=\frac{2}{\sqrt{5}}$
For $x$ and $y$, we have
$x=X \cos \theta-Y \sin \theta=\frac{X-2 Y}{\sqrt{5}}$
and $y=X \sin \theta+Y \cos \theta=\frac{2 X+Y}{\sqrt{5}}$
The equation $4 x y-3 x^{2}=a^{2}$ reduces to
$\frac{4(X-2 Y)}{\sqrt{5}} \cdot \frac{(2X+Y)}{\sqrt{5}}-3\left(\frac{X-2 Y}{\sqrt{5}}\right)^{2}=a^{2}$
$\Rightarrow 4\left(2 X^{2}-2 Y^{2}-3 X Y\right)-3\left(X^{2}-4 X Y+4 Y^{2}\right)=5 a^{2}$
$\Rightarrow 5 X^{2}-20 Y^{2}=5 a^{2}$
$\Rightarrow X^{2}-4 Y^{2}=a^{2}$