The lines L_1: y-x=0 and L_2: 2x+y=0 intersec | vedclass

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(C) Step $1$: Find the intersection points $P$ and $Q$.For $L_1: y=x$ and $L_3: y=-2$,we get $P = (-2, -2)$.For $L_2: y=-2x$ and $L_3: y=-2$,we get $-

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Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is a correct explanation for Statement-$I$

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(C) Step $1$: Find the intersection points $P$ and $Q$.For $L_1: y=x$ and $L_3: y=-2$,we get $P = (-2, -2)$.For $L_2: y=-2x$ and $L_3: y=-2$,we get $-2 = -2x \implies x=1$,so $Q = (1, -2)$.Step $2$: Calculate lengths $PR$ and $RQ$ using the Angle Bisector Theorem.The distance $OP = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}$.The distance $OQ = \sqrt{1^2 + (-2)^2} = \sqrt{5}$.The origin $O(0,0)$ is the intersection of $L_1$ and $L_2$. Since the bisector of $\angle POQ$ divides $PQ$ at $R$,by the Angle Bisector Theorem,$PR:RQ = OP:OQ = 2\sqrt{2}:\sqrt{5}$.Thus,Statement-$I$ is true.Step $3$: Evaluate Statement-$II$.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. Statement-$II$ is a standard geometric theorem. Thus,Statement-$II$ is true and provides the correct explanation for Statement-$I$.

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Statement $-I$: Two lines which pass through a given fixed point and are equally inclined to two other lines passing through the same point,are always perpendicular to each other.
Statement $-II$: Angle bisectors of two intersecting lines are always perpendicular to each other.

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