Question

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

• A. $X^2-4Y^2=a^2$
• B. $X^2+4Y^2=a^2$
• C. $X^2+4Y^2=-a^2$
• 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
$x=X\cos \theta -Y\sin \theta =\frac{X-2Y}{\sqrt{5}}$
and $y=X\sin \theta +Y\cos \theta =\frac{2X+Y}{\sqrt{5}}$
The equation $4xy-3x^2=a^2$ reduces to
$\frac{4(X-2Y)}{\sqrt{5}}\cdot \frac{(2X+Y)}{\sqrt{5}}-3{\left(\frac{X-2Y}{\sqrt{5}}\right)}^2=a^2$
$\Rightarrow 4\left(2X^2-2Y^2-3XY\right)-3\left(X^2-4XY+4Y^2\right)=5a^2$
$\Rightarrow 5X^2-20Y^2=5a^2$
$\Rightarrow X^2-4Y^2=a^2$