Question

The lines \( L_1: y - x = 0 \) and \( L_2: 2x + y = 0 \) intersect the line \( L_3: y + 2 = 0 \) at points \( P \) and \( Q \) respectively. The bisector of the acute angle between \( L_1 \) and \( L_2 \) intersects \( L_3 \) at \( R \). Statement-1: \( PR : RQ = 2\sqrt{2} : \sqrt{5} \) Statement-2: In any triangle, the bisector of an angle divides the triangle into two similar triangles.

• A. Statement-1 is true, Statement-2 is false
• B. Statement-1 is false, Statement-2 is true
• C. Statement-1 and Statement-2 are both true
• D. Statement-1 and Statement-2 are both false

Hint:

The angle bisector divides the opposite side in the ratio of adjacent sides — not necessarily creating similar triangles unless it's an isosceles setup.

Answer 1

The Correct Option is A

- Find intersections \( P \) and \( Q \) with line \( y = -2 \): - \( L_1 \): \( y = x \Rightarrow x = -2 \Rightarrow P = (-2, -2) \) - \( L_2 \): \( 2x + y = 0 \Rightarrow 2x = 2 \Rightarrow x = -1, Q = (-1, -2) \) - Acute angle bisector of \( L_1 \) and \( L_2 \) intersects the x-axis at a ratio determined by angle bisector theorem and direction ratios. Calculating that leads to a segment on \( L_3 \) dividing \( PR \) and \( RQ \) in ratio \( 2\sqrt{2} : \sqrt{5} \) - Statement 2 is incorrect as angle bisectors in general do not guarantee similar triangles in arbitrary configurations.