Question

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

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

Answer 1

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

From $PA=PB$, we get

$(x-3{)}^2+(y-4{)}^2=(x-5{)}^2+(y+2{)}^2$

$-6x+9-8y+16=-10x+25+4y+4$

$x=3y+1$ is the line on which P lies

Equation of line AB, $y-4=\frac{-2-4}{5-3}(x-3)$

$y=-3x+13$

$AB=2\sqrt{10}$

Hence, height = $\sqrt{10}$

Perpendicular length from P on AB, $\frac{|y+3x-13|}{\sqrt{10}}=\sqrt{10}$

$|y+3x-13|=10$

$|y+9y+3-13|=10$

$|10y-10|=10$

which is possible only for $y=0,2$

Equivalently, $x=1,7$

So co-ordinates of P can be either $(7,2)or(1,0)$