Lines L_1: y - x = 0 and L_2: 2x + y = 0 inte | vedclass

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(A) For $L_1: y = x$ and $L_3: y = -2$,the intersection point $P$ is $(-2, -2)$.For $L_2: y = -2x$ and $L_3: y = -2$,the intersection point $Q$ is $(1

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Statement-$1$ and Statement-$2$ are both true and Statement-$2$ is the correct explanation of Statement-$1$.

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(A) For $L_1: y = x$ and $L_3: y = -2$,the intersection point $P$ is $(-2, -2)$.For $L_2: y = -2x$ and $L_3: y = -2$,the intersection point $Q$ is $(1, -2)$.The distance $PQ = |1 - (-2)| = 3$.The angle bisector theorem states that the bisector of an angle of a triangle divides the opposite side into segments proportional to the other two sides.Let $O$ be the origin $(0,0)$. The lengths of the sides are $OP = \sqrt{(-2)^2 + (-2)^2} = 2\sqrt{2}$ and $OQ = \sqrt{1^2 + (-2)^2} = \sqrt{5}$.The bisector of $\angle POQ$ divides $PQ$ in the ratio $OP:OQ = 2\sqrt{2} : \sqrt{5}$.Thus,$PR:RQ = 2\sqrt{2} : \sqrt{5}$.Statement-$1$ is true.Statement-$2$ is the Angle Bisector Theorem,which is a standard geometric property,and it is the correct explanation for the ratio calculation in Statement-$1$.

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