Search any question & find its solution
Question:
Answered & Verified by Expert
$\mathrm{A}(3,4)$ and $\mathrm{B}(5,-2)$ are two points and $\mathrm{P}$ is a point such that $\mathrm{PA}=\mathrm{PB}$. If the area of triangle $\mathrm{PAB}$ is 10 square unit, what are the coordinates of P ?
Options:
Solution:
1576 Upvotes
Verified Answer
The correct answer is:
$(1,0)$ or $(7,2)$
Given $\mathrm{A}(3,4)$ and $\mathrm{B}(5,-2)$ Let, $\mathrm{P}(\mathrm{x}, \mathrm{y})$
Given that, $\mathrm{PA}=\mathrm{PB}$ $\Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2}$
$\Rightarrow(x-3)^{2}+(y-4)^{2}=(x-5)^{2}+(y+2)^{2}$
$\Rightarrow x^{2}-6 x+9+y^{2}-8 y+16$
$=x^{2}-10 x+25+y^{2}+4 y+4$
$\Rightarrow 4 x-12 y=4$
$\Rightarrow x-3 y=1$ ...(i)
$\therefore$ Area of $\Delta \mathrm{PAB}=10$
$\frac{1}{2}\left|\begin{array}{lll}x & y & 1 \\ 3 & 4 & 1 \\ 5 & -2 & 1\end{array}\right|=\pm 10$
$\Rightarrow x(4+2)-y(3-5)+1(-6-20)=\pm 20$
$\Rightarrow 6 \mathrm{x}+2 \mathrm{y}-26=\pm 20$
$\Rightarrow 6 \mathrm{x}+2 \mathrm{y}-26=20$
or, $6 x+2 y-26=-20$
$\Rightarrow 6 x+2 y=46$...(ii)
or $6 x+2 y=6$...(iii)
From eqs. (i) and (ii), we get $x=7, y=2$
Similarly, from eqs. (i) and (iii), we get $\mathrm{x}=1, \mathrm{y}=0$
Hence, coordinates of $\mathrm{P}$ are $(7,2)$ or $(1,0)$
Given that, $\mathrm{PA}=\mathrm{PB}$ $\Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2}$
$\Rightarrow(x-3)^{2}+(y-4)^{2}=(x-5)^{2}+(y+2)^{2}$
$\Rightarrow x^{2}-6 x+9+y^{2}-8 y+16$
$=x^{2}-10 x+25+y^{2}+4 y+4$
$\Rightarrow 4 x-12 y=4$
$\Rightarrow x-3 y=1$ ...(i)
$\therefore$ Area of $\Delta \mathrm{PAB}=10$
$\frac{1}{2}\left|\begin{array}{lll}x & y & 1 \\ 3 & 4 & 1 \\ 5 & -2 & 1\end{array}\right|=\pm 10$
$\Rightarrow x(4+2)-y(3-5)+1(-6-20)=\pm 20$
$\Rightarrow 6 \mathrm{x}+2 \mathrm{y}-26=\pm 20$
$\Rightarrow 6 \mathrm{x}+2 \mathrm{y}-26=20$
or, $6 x+2 y-26=-20$
$\Rightarrow 6 x+2 y=46$...(ii)
or $6 x+2 y=6$...(iii)
From eqs. (i) and (ii), we get $x=7, y=2$
Similarly, from eqs. (i) and (iii), we get $\mathrm{x}=1, \mathrm{y}=0$
Hence, coordinates of $\mathrm{P}$ are $(7,2)$ or $(1,0)$
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.