Question

The line $L_1:y-x=0$ 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-l:
The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$
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 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
• D. Statement-1 is false, Statement-2 is true.

Answer 1

The correct option is C Statement-1 is true, Statement-2 is false

Intersection of $L_1$ and $L_3$ is $P=(-2,-2)$
Intersection of $L_2$ and $L_3$ is $Q=(1,-2)$
Now, Intersection of $L_1$ and $L_2$ is $O=(0,0)$
Equation of angular bisector of $\Delta OPQ$ will be $(\sqrt{5}+2\sqrt{2})x=(\sqrt{5}-\sqrt{2})y$
In $\Delta OPQ$, angle bisector of $O$ divides $PQ$ in the ratio of $OP:OQ$ which is $2\sqrt{2}:\sqrt{5}$ but it does not divide triangle into two similar triangle.
Hence, option 'C' is correct.