Question

Assertion :

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 $R$.

The ratio $PR:RQ$ equal $2\sqrt{2}:\sqrt{5}$ Reason: In nay triangle, bisector of an angle divides the triangle into two similar triangles.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Points of intersection of $L_1$ and $L_2.$ i.e., $A$ is $(0,0).$

Points of intersection $L_2$ and $L_3$ is $Q(1,-2)$ and that of $L_1$ and $L_3$ is$P{(}_2,-2).$

Now, $AP=\sqrt{{(-2,-0)}^2+{(-2,-0)}^2}=2\sqrt{2}$

and $AQ=\sqrt{{(1-0)}^2+{(-2-0)}^2}=\sqrt{5}$

By angle bisector theorem $RP:RQ=AP:AQ=2\sqrt{2}:\sqrt{5}$

$\therefore$ Assertion is correct but reason is incorrect.