Q84 / 160 · TS EAMCET 2021 (05 Aug Shift 2) · Mathematics

The angle by which axes are to be rotated without changing the origin so that the transformed equation of $x^{2} + 4 x y - y^{2} = 0$ in new coordinates $(X, Y)$ does not contain $X Y$ term is

Solution

Here, we have the equation,

$x^{2} + 4 x y - y^{2} = 0$

$\Rightarrow A = 1, B = 4, C = - 1$

$\Rightarrow c o t 2 θ = \frac{A - C}{B} = \frac{1 - (- 1)}{4} = \frac{1}{2}$

$\Rightarrow \tan 2 θ = 2 \Rightarrow 2 θ = \tan^{- 1} (2) \Rightarrow θ = \frac{1}{2} \tan^{- 1} (2)$

So, if we rotate the axes with $θ = \frac{1}{2} \tan^{- 1} (2)$ ,

$x y -$ term will be eliminated.

Hence, the correct answer is option: 'A'.