IIT-JEE 2007 · Paper 2 · Mathematics Q57

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MathematicsAssertion and reason+3 / -1Question 57 of 66

Lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$ , respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$ .

STATEMENT-1: The ratio $PR:RQ$ equals $2\sqrt2:\sqrt5$ .

because

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

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

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

Statement-1 is True, Statement-2 is False

Statement-1 is False, Statement-2 is True

Answer and worked solution

Official answer: C

Points. $L_3$ : $y=-2$ . $P=(-2,-2)$ on $L_1$ and $Q=(1,-2)$ on $L_2$ . Both lines pass through the origin O.

The angle at O. $\cos\angle POQ=\dfrac{(-2)(1)+(-2)(-2)}{2\sqrt2\cdot\sqrt5}=\dfrac1{\sqrt{10}}>0$ , so $\angle POQ$ is the acute angle and its bisector is the internal bisector of triangle OPQ.

Statement 1. By the angle-bisector theorem $PR:RQ=OP:OQ=2\sqrt2:\sqrt5$ . True.

Statement 2. An angle bisector does not in general split a triangle into similar triangles. False.

(C)

Topics:Angle between lines Family of lines and pair of lines

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IIT-JEE 2007 · Paper 2, Mathematics question 57