TS EAMCET · Maths · Straight Lines

If $θ_{1}, θ_{2}, θ_{3}$ are respectively the angles by which the coordinate axes are to be rotated to eliminate the $x y$ term from the following equations, then the descending order of these angles is $A_{1} = 3 x^{2} + 5 x y + 3 y^{2} + 2 x + 3 y + 4 = 0$
$A_{2} = 5 x^{2} + 2 \sqrt{3} x y + 3 y^{2} + 6 = 0$ $A_{3} = 4 x^{2} + \sqrt{3} x y + 5 y^{2} - 4 = 0$

A $θ_{1}, θ_{2}, θ_{3}$
B $θ_{3}, θ_{1}, θ_{2}$
C $θ_{2}, θ_{1}, θ_{3}$
D $θ_{3}, θ_{2}, θ_{1}$

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Answer & Solution

Correct Answer

(B) $θ_{3}, θ_{1}, θ_{2}$

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Detailed explanation

For A1 ⇒3x'cosθ1-y'sinθ12+5x'cosθ1-y'sinθ1x'sinθ1+y'cosθ1+3x'sinθ1+y'cosθ12+2x'cosθ1-y'sinθ1+3x'sinθ1+y'cosθ1+4=0 Now, the coefficient of x'y'=-3sin2θ1+5cos2θ1+3sin2θ1=0 ⇒θ1=π4 For…

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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If θ1,θ2,θ3 are respectively the angles by which the coordinate axes are to be rotated to eliminate the xy term from the following equations, then the descending order of these angles is A1=3x2+5xy+3y2+2x+3y+4=0A2=5x2+23xy+3y2+6=0 A3=4x2+3xy+5y2-4=0

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