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.
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.
Show 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.
Updated On: May 13, 2025
- Statement-1 is true, Statement-2 is false
- Statement-1 is false, Statement-2 is true
- Statement-1 and Statement-2 are both true
- Statement-1 and Statement-2 are both false
Show Solution
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The Correct Option is A
Solution and Explanation
- 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.
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