Question

$A[3,4]$ and $B[5,-2]$ are to given points. If $PA=PB$ and area of triangle $PAB=10$, Then $P$ is?

Answer 1

$A(3,4)$ and $B(5,-2)$

$PA=PB$

Area of triangle $PAB=10$

Let the point $P$ be $(x,y)$

$PA=\sqrt{(3-x{)}^2+(4-y{)}^2}$

$PB=\sqrt{(5-x{)}^2+(-2-y{)}^2}$

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

squaring on both sides

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

$\Rightarrow 9-6x+x^2+16-8y+y^2=25-10x+x^2+4+4y+y^2$

$\Rightarrow 25-6x-8y=29-10x+4y$

$\Rightarrow 4x-12y=4$

$\Rightarrow x-3y=1$ (Equation1)

area of triangle

$=\frac{1}{2}\left[x_1\right(y_2-y_1)+x_2(y_3-y_3)+x_3(y_1-y_2\left)\right]$

$=\frac{1}{2}\left[3\right(-2-y)+5(y-4)+x(4+2\left)\right]=10$

$=-6-3y+5y-20+4x+2x=20$

$=-26+2y+6x=20$

$=3x+y=23$ (Equation2)

From equation1

$x=1+3y$

put $x$ in equation2

$3(1+3y)+y=23$

$\Rightarrow 3+9y+y=23$

$\Rightarrow 3+10y=23$

$\Rightarrow 10y=20$

$\Rightarrow y=2$

$\therefore 3x+y=23$

$\Rightarrow 3x+2=23$

$\Rightarrow 3x=21$

$\Rightarrow x=7$

So , the coordinate of P $=(7,2)$