Search any question & find its solution
Question:
Answered & Verified by Expert
Let $\mathrm{PQR}$ be a right angled isosceles triangle, right angled at $\mathrm{P}(2,1)$. If the equation of the line $\mathrm{QR}$ is $2 x+y=3$, then the equation representing the pair of lines $P Q$ and $P R$ is
Options:
Solution:
2980 Upvotes
Verified Answer
The correct answer is:
$3 x^2-3 y^2+8 x y-20 x-10 y+25=0$
$\begin{array}{ll} & \text { Slope of } Q R=-2 . \\ & \text { Slope of } P Q=\mathrm{m}_1 \\ \therefore \quad & \tan 45^{\circ}=\left|\frac{\mathrm{m}_1+2}{1+\mathrm{m}_1(-2)}\right| \\ \Rightarrow & 1=\left|\frac{\mathrm{m}_1+2}{1-2 \mathrm{~m}_1}\right| \\ \Rightarrow & \mathrm{m}_1=-\frac{1}{3}\end{array}$

$\therefore \quad$ Equation of $\mathrm{PQ}$ passing through point $\mathrm{P}(2,1)$ and having slope $\frac{-1}{3}$ is
$$
\begin{aligned}
& y-1=-\frac{1}{3}(x-2) \\
& \Rightarrow 3(y-1)+(x-2)=0
\end{aligned}
$$
Slope of $\mathrm{PR}=\mathrm{m}_2=3$
$\therefore \quad$ equation of $\mathrm{PR}$ is
$$
\begin{aligned}
& y-1=3(x-2) \\
& \Rightarrow(y-1)-3(x-2)=0
\end{aligned}
$$
$\therefore \quad$ The joint equation of the lines is
$$
\begin{aligned}
& {[3(y-1)+(x-2)][(y-1)-3(x-2)]=0} \\
& \Rightarrow 3(y-1)^2-8(y-1)(x-2)-3(x-2)^2=0 \\
& \Rightarrow 3\left(x^2-4 x+4\right)+8(x y-x-2 y+2) \\
& \quad-3\left(y^2-2 y+1\right)=0 \\
& \Rightarrow 3 x^2-3 y^2+8 x y-20 x-10 y+25=0
\end{aligned}
$$

$\therefore \quad$ Equation of $\mathrm{PQ}$ passing through point $\mathrm{P}(2,1)$ and having slope $\frac{-1}{3}$ is
$$
\begin{aligned}
& y-1=-\frac{1}{3}(x-2) \\
& \Rightarrow 3(y-1)+(x-2)=0
\end{aligned}
$$
Slope of $\mathrm{PR}=\mathrm{m}_2=3$
$\therefore \quad$ equation of $\mathrm{PR}$ is
$$
\begin{aligned}
& y-1=3(x-2) \\
& \Rightarrow(y-1)-3(x-2)=0
\end{aligned}
$$
$\therefore \quad$ The joint equation of the lines is
$$
\begin{aligned}
& {[3(y-1)+(x-2)][(y-1)-3(x-2)]=0} \\
& \Rightarrow 3(y-1)^2-8(y-1)(x-2)-3(x-2)^2=0 \\
& \Rightarrow 3\left(x^2-4 x+4\right)+8(x y-x-2 y+2) \\
& \quad-3\left(y^2-2 y+1\right)=0 \\
& \Rightarrow 3 x^2-3 y^2+8 x y-20 x-10 y+25=0
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.