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 angle between \(L_1\) and \(L_2\) divides the segment \(PQ\) internally at \(R\). Consider: Statement-I: \(PR : RQ = 2\sqrt{2} : \sqrt{5}\).
Statement-II: In any triangle, bisector of an angle divides that triangle into two similar triangles.
Which statement(s) is/are correct?

• A. Statement-I is true, Statement-II is false
• B. Statement-I is false, Statement-II is true
• C. Statement-I and Statement-II are true, but Statement-II is not a correct explanation for Statement-I

Hint:

Use geometric properties of angle bisectors and coordinate geometry to verify ratios and similarity.

Answer 1

The Correct Option is A

Calculate \(P\) and \(Q\) from intersections: \(P=(-2,-2)\), \(Q=(1,-2)\). Bisector divides \(PQ\) internally in ratio \(2\sqrt{2} : \sqrt{5}\), so Statement-I true. Statement-II is false; angle bisector divides opposite side in ratio of adjacent sides, but does not always form similar triangles.