JEE Challenger
More from Straight Lines
MathMedAdv 2007 • Paper 2

Angle Bisector Ratio in Coordinate Geometry

This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.

Lines L1:y−x=0L_1 : y - x = 0 and L2:2x+y=0L_2 : 2x + y = 0 intersect the line L3:y+2=0L_3 : y + 2 = 0 at PP and QQ, respectively. The bisector of the acute angle between L1L_1 and L2L_2 intersects L3L_3 at RR.

STATEMENT-1 :\textbf{STATEMENT-1 :} The ratio PR:RQPR : RQ equals 22:52\sqrt{2} : \sqrt{5}.

because\textbf{because}

STATEMENT-2 :\textbf{STATEMENT-2 :} In any triangle, bisector of an angle divides the triangle into two similar triangles.

Options

A

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False

Correct
D

Statement-1 is False, Statement-2 is True

Topics & Concepts

Straight LinesAngle BisectorProperties of TrianglesRatio of Segments

Step-by-Step Solution

  1. Coordinates of Vertices:
  • Both L1L_1 and L2L_2 pass through the origin O(0,0)O(0, 0).
  • Point PP: intersection of y−x=0y - x = 0 with y=−2  ⟹  x=−2  ⟹  P(−2,−2)y = -2 \implies x = -2 \implies P(-2, -2).
  • Point QQ: intersection of 2x+y=02x + y = 0 with y=−2  ⟹  2x−2=0  ⟹  x=1  ⟹  Q(1,−2)y = -2 \implies 2x - 2 = 0 \implies x = 1 \implies Q(1, -2).

  1. Check Statement-1:
    In △OPQ\triangle OPQ, ray OROR is an internal angle bisector of ∠POQ\angle POQ meeting the opposite side PQPQ at RR. By the Angle Bisector Theorem:
    PRRQ=OPOQ\frac{PR}{RQ} = \frac{OP}{OQ}
    Calculating lengths:
    OP=(−2)2+(−2)2=8=22OP = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}
    OQ=12+(−2)2=5OQ = \sqrt{1^2 + (-2)^2} = \sqrt{5}
    Thus, PRRQ=225\frac{PR}{RQ} = \frac{2\sqrt{2}}{\sqrt{5}}, meaning PR:RQ=22:5PR : RQ = 2\sqrt{2} : \sqrt{5}. Statement-1 is True.

  2. Check Statement-2:
    An angle bisector divides a triangle into two triangles with equal angles at the bisected vertex, but their other angles are generally not equal, so the two triangles are not similar (they are only similar in an isosceles triangle where the bisector is also an altitude and median, dividing it into congruent triangles). Hence, Statement-2 is False.

Correct Answer: C

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