Question

If \[ x \cos \theta + y \sin \theta = 5, \quad x \sin \theta - y \cos \theta = 3, \quad \text{then the value of} \, x^2 + y^2 \, \text{is} \]

• A. 17
• B. 8
• C. 12
• D. 34

Hint:

When given two equations with trigonometric terms, square and add them to eliminate the cross-product terms and find \( x^2 + y^2 \).

Answer 1

The Correct Option is D

Step 1: Squaring both equations.
We are given two equations: \[ x \cos \theta + y \sin \theta = 5 \quad \text{(1)} \] \[ x \sin \theta - y \cos \theta = 3 \quad \text{(2)} \] Squaring both equations: \[ (x \cos \theta + y \sin \theta)^2 = 25 \] \[ (x \sin \theta - y \cos \theta)^2 = 9 \]
Step 2: Adding the squared equations.
Now add the two equations: \[ (x^2 \cos^2 \theta + 2xy \cos \theta \sin \theta + y^2 \sin^2 \theta) + (x^2 \sin^2 \theta - 2xy \cos \theta \sin \theta + y^2 \cos^2 \theta) = 25 + 9 \] Simplifying this, we get: \[ x^2 (\cos^2 \theta + \sin^2 \theta) + y^2 (\cos^2 \theta + \sin^2 \theta) = 34 \] Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we get: \[ x^2 + y^2 = 34 \]
Step 3: Conclusion.
Thus, \( x^2 + y^2 = 34 \), which makes option (D) the correct answer.