Question

If the coordinates of two points A and B are $(3,4)$ and $(5,-2)$, respectively, then the coordinates of any point P if $PA=PB$ and area of $\Delta PAB=10$ is

• A. $(7,2)$ or $(1,0)$
• B. $(-7,2)$ or $(3,0)$
• C. $(7,-2)$ or $(5,0)$
• D. $(7,-2)$ or $(-1,0)$

Answer 1

The correct option is A $(7,2)$ or $(1,0)$

Let P $=(x,y)$
Given, $PA=PB$
$=>{PA}^2={PB}^2$
$=>{(x-3)}^2+{(y-4)}^2={(x-5)}^2+{(y+2)}^2$
$=>x^2-6x+9+y^2-8y+16=x^2-10x+25+y^2+4y+4$
$=>4x-12y=4$
$=>x-3y=1$ .......(i)
Also, Area of $\Delta PAB=10$
$\left|\frac{x(4+2)+3(-2-y)+5(y-4)}{2}\right|=10$
$\left|\frac{6x-6-2y+5y-20}{2}\right|=10$
$\frac{6x+3y-26}{2}=\pm 10$
$6x+2y-26=\pm 20$
$6x+2y=46$ ........(ii)
or $6x+2y=6$ .......(iii)
Solving equation (i) and (ii), we get $x=7,y=2$

Solving equation (i) and (iii), we get $x=1,y=0$.
So, the co-ordinates of P are $(7,2)$ or $(1,0)$.