TS EAMCET · Maths · Straight Lines

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

A $\frac{1}{2} \tan^{- 1} (2)$
B $\tan^{- 1} (2)$
C $\frac{π}{8}$
D $\frac{π}{4}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{1}{2} \tan^{- 1} (2)$

Step-by-step Solution

Detailed explanation

Here, we have the equation, x2+4xy-y2=0 ⇒A=1, B=4, C=-1 ⇒cot2θ=A-CB=1--14=12 ⇒tan 2θ=2 ⇒ 2θ=tan-12 ⇒ θ=12tan-12 So, if we rotate the axes with θ=12tan-12, xy-term will be eliminated.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The angle by which axes are to be rotated without changing the origin so that the transformed equation of x2+4xy-y2=0 in new coordinates X, Y does not contain XY term is

Step-by-step Solution

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