Let $\mathrm{PQR}$ be a right angled isosceles triangle, right angled at $P(2,1)$. If the equation of the side $Q R$ is $2 x+y=3$, then the equation of one of the sides other than $Q R$ is

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Aicharya · Accepted Answer

Since $	riangle P Q R$ is isosceles right angled triangle. Hence $\angle P Q R=\angle P R Q=45^{\circ}$ Let $m_1$ and $m_2$ be the slope of line $P Q$ and line $P R$ respectively. Given line $Q R$ is $2 x+y=3$ $\Rightarrow y=-2 x+3$. Hence slope of $Q R=m=-2$ Since the angle between $P Q$ and $R Q$ is $45^{\circ}$. Hence $ \begin{aligned} & 	an 45^{\circ}=\frac{m_1-m}{1+m_1 m}=\frac{m_1+2}{1-2 m_1} \ & \Rightarrow m_1+2=1-2 m_1 \Rightarrow 3 m_1=-1 \ & \Rightarrow m_1=-\frac{1}{3} \end{aligned} $ Since $P R$ and $P Q$ are perpendicular lines Hence $m_1 m_2=-1$ $ \Rightarrow m_2=-\frac{1}{m_1}=3 \Rightarrow m_2=3 $ Hence equation of line $P Q$, $ \begin{aligned} & y-1=m_1(x-2)=\left(-\frac{1}{3}ight)(x-2) \ & \Rightarrow 3 y+x+1=0 \end{aligned} $ Hence equation of line $P R$, $ \begin{aligned} & \Rightarrow y-1=m_2(x-2)=3(x-2) \ & \Rightarrow 3 x-y-5=0 \end{aligned} $

Let $\mathrm{PQR}$ be a right angled isosceles triangle, right angled at $P(2,1)$. If the equation of the…

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