The 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 \(L_{2}\) intersect \(L_{3}\) at \(R\). Statement -1 : The ratio \(\mathrm{PR}: \mathrm{RQ}\) equals \(2 \sqrt{2}: \sqrt{5}\). Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

(B) Statement -1 is true, Statement- 2 is false.

(A) Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement - 1

(C) Statement -1 is false, Statement- 2 is true.

(D) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1

JEE Main · 2011 · 10 Apr 2011 · Offline · Straight Lines

The 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}

intersect

L_{3}

R

.
Statement -1 : The ratio

PR : RQ

equals

22 : 5

.
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 not a correct explanation for Statement - 1
(B)Statement -1 is true, Statement- 2 is false.✓ Correct
(C)Statement -1 is false, Statement- 2 is true.
(D)Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1

Answer: (B)

Solution

The line

L_{1} : y = x

meets

L_{3} : y = - 2

P (- 2, - 2)

, and

L_{2} : y = - 2 x

meets it at

Q (1, - 2)

. The bisector of the acute angle between

L_{1}

and

L_{2}

passes through the origin and meets

L_{3}

R

; working out that bisector and its intersection gives the division ratio

P R : R Q = 22 : 5

, so the first statement is true. The second statement, that an angle bisector splits a triangle into two similar triangles, is false in general, so here the first statement is true and the second is false.

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option

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JEE Main · 2011 · 10 Apr 2011 · Offline · Straight Lines

Solution

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