Question

If the lines given by \( (x^2 + y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2 \) are perpendicular to each other, then \( \sin^2 \alpha + \tan^2 \alpha = \):

• A. \( \frac{15}{4} \)
• B. \( 0 \)
• C. \( \frac{3}{2} \)
• D. \( \frac{7}{12} \)

Hint:

In problems involving perpendicular lines, remember to use the slope condition and simplify carefully.

Answer 1

The Correct Option is C

We are given the equation \( (x^2 + y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2 \) for two perpendicular lines.
Step 1: Simplify the given equation. Expand the right-hand side: \[ (x \cos \alpha - y \sin \alpha)^2 = x^2 \cos^2 \alpha - 2xy \cos \alpha \sin \alpha + y^2 \sin^2 \alpha \] Substitute this into the original equation: \[ (x^2 + y^2) \sin^2 \alpha = x^2 \cos^2 \alpha - 2xy \cos \alpha \sin \alpha + y^2 \sin^2 \alpha \] Now, simplify the equation. Since the lines are perpendicular, we use the condition for perpendicular lines in terms of slopes to determine that \( \sin^2 \alpha + \tan^2 \alpha = \frac{3}{2} \).