SATHEE: Straight Line and Pair of Straight Lines 2 Question 4

Straight Line and Pair of Straight Lines 2 Question 4

4. 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 I The ratio $P R : R Q$ equals $2 \sqrt{2} : \sqrt{5}$ .

Because

Statement II In any triangle, bisector of an angle divides the triangle into two similar triangles. (2007, 3M)

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Answer:

Correct Answer: 4. (c)

Solution:

It is not necessary that the bisector of an angle will divide the triangle into two similar triangles, therefore, statement II is false.

Now, we verify Statement I.

$△ O P Q, O R$ is the internal bisector of $∠ P O Q$ .

$\begin{array}{lll} ∴ & \frac{P R}{R Q} = \frac{O P}{O Q} \\ \Rightarrow & \frac{P R}{R Q} = \frac{\sqrt{2^{2} + 2^{2}}}{\sqrt{1^{2} + 2^{2}}} = \frac{2 \sqrt{2}}{\sqrt{5}} \end{array}$

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Straight Line and Pair of Straight Lines 2 Question 4

4. Lines L1:y−x=0 and L2:2x+y=0 intersect the line L3:y+2=0 at P and Q, respectively. The bisector of the acute angle between L1 and L2 intersects L3 at R.

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4. 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 $R$ .