Question

If the co-ordinates of two points A and B are (3, 4) and (5, -2) respectively then the co-ordinates 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)

or, ${PA}^2={PB}^2$
$=>{(x-3)}^2+{(y-4)}^2={(x-5)}^2+{(y+2)}^2$
$x^2-6x+9+y^2-8y+16=x^2-10x+25+y^2+4y+4$
$4x-12y=4$
$x-3y=1$ .......(i)
Also Area of $PAB=10$
$|\frac{\left(x\right(4+2)+3(-2-y)+5(y-4)}{2}=10$
$\left|\frac{6x-6-2y+5y-20}{2}\right|=10$
$\frac{6x+3y-26}{2}=\pm 10$
$ 6x+2y26=
\pm 20 $
$ 6x+2y =46 $........\left(ii\right)or$ 6x + 2y = 6 $.......\left(iii\right)Solvingequation\left(i\right)and\left(ii\right)$ x = 7, y = 2 $ and solving equation (i) and (iii) $x=1,y=0$

So, the co-ordinates of P are $(7,2)or(1,0)$