Lines $L_{1}: y-x=0$ and $L_{2}: 2 x+y=0$ 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 $2 \sqrt{2}: \sqrt{5}$. because STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

(C) Statement-1 is True, Statement-2 is False

(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

(D) Statement-1 is False, Statement-2 is True

JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Straight Lines

Lines

L_{1} : y - x = 0

and

L_{2} : 2 x + y = 0

intersect the line

L_{3} : y + 2 = 0

P

and

Q

, respectively. The bisector of the acute angle between

L_{1}

and

L_{2}

intersects

L_{3}

R

.

STATEMENT-1 : The ratio

P R : R Q

equals

22 : 5

.

because

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

(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

Answer: (C)

Solution

Finding intersection points:

$L_{1} : y = x$ and $L_{3} : y = - 2$ gives $P = (- 2, - 2)$
$L_{2} : y = - 2 x$ and $L_{3} : y = - 2$ gives $Q = (1, - 2)$

The lines

L_{1}

and

L_{2}

intersect at origin

O = (0, 0)

.

In triangle

O P Q

, by the angle bisector theorem:

\frac{P R}{R Q} = \frac{O P}{O Q}

Calculating distances:

$O P = (- 2)^{2} + (- 2)^{2} = 22$
$O Q = 1^{2} + (- 2)^{2} = 5$

Therefore,

\frac{P R}{R Q} = \frac{2 2}{5}

, so Statement-1 is TRUE.

Statement-2 is FALSE. The angle bisector theorem states that the bisector divides the opposite side in the ratio of the adjacent sides, but it does NOT create two similar triangles.

JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Single correct option

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JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Straight Lines

Solution

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