Question

Assertion :$L$ines $L_1$ : $y-x$ $=0$ 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-$I$ : The ratio $PR$: $RQ$ equals $2\sqrt{2}$ : $\sqrt{5}$. Reason: 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

$L_1:y=x$ ........ $\left(i\right)$

$L_2:y=-2x$ ........ $\left(ii\right)$

$L_3:y=-2$ ....... $\left(iii\right)$

Then, $P=(-2,-2)$ and $Q=(1,-2)$

The slopes of first two lines are $m_1=1$ and $m_2=-2$

$\tan \frac{\theta }{2}=\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$\left|\frac{3}{-1}\right|=3$

Let $\tan \frac{\theta }{2}=x$

$\frac{2x}{1-x^2}=3$

$2x=3-3x^2$

$3x^2+2x-3=0$

$x=\frac{-2\pm \sqrt{4+36}}{6}$

$x=\frac{-2\pm 2\sqrt{10}}{6}=\frac{-1\pm \sqrt{10}}{3}$

Now since $L_4$ is the angle bisector

$m_4=\tan (\frac{\theta }{2}+45°)=\frac{\frac{-1+\sqrt{10}}{3}+1}{1-\frac{\sqrt{10}-1}{3}}=\frac{2+\sqrt{10}}{4-\sqrt{10}}$

$R=(\frac{-2(4-\sqrt{10})}{(2+\sqrt{10})},-2)$

$PR=\frac{-2(4-\sqrt{10})}{2+\sqrt{10}}+\left(2\right)=\frac{4+2\sqrt{10}-8+2\sqrt{10}}{2+\sqrt{10}}=\frac{-4+4\sqrt{10}}{2+\sqrt{10}}$

$PR=1+\frac{2(4-\sqrt{10})}{2+\sqrt{10}}=\frac{10-\sqrt{10}}{2+\sqrt{10}}$

$\frac{PR}{RQ}=\frac{4(\sqrt{10}-1)}{(10-\sqrt{10})}=\frac{4(\sqrt{10}-1)}{\sqrt{10}(\sqrt{10}-1)}=\frac{2\sqrt{2}}{\sqrt{5}}$

Statement $1$ is correct.

Statement $-2:$

In any triangle, bisector of an angle divides the triangle into two similar triangle. This is wrong.

As in $△ABC,$

If $AD$ bisects $\angle A$

then, $\angle \frac{A}{2}=\angle \frac{A}{2}$

But we can't conclude about other two angles and the side.