Two points A and B with co-ordinates ( 1 , 1 ) and ( - 2 , 3 ) are given respectively. Then, the locus of a…

Given,

$A(1,1) and B(-2,3)$

$Let P(x,y)$ be the point as per given condition

$A(△PAB)=9$

We know that, if 3 points are given then the area of triangle is given by -
$\frac{1}{2}\left|\begin{array}{ccc}x& y& 1\\ 1& 1& 1\\ -2& 3& 1\end{array}\right|=9$

$\Rightarrow \left|x(1-3)-y(1+2)+1(3+2)\right|=9\times 2$

$\Rightarrow \left|x(-2)-3y+5\right|=18$

$\Rightarrow -2x-3y+5=\pm 18$

$\Rightarrow -2x-3y=\pm 18-5$

$\Rightarrow -2x-3y=18-5 and -2x-3y=-18-5$

$\Rightarrow -2x-3y=13 and -2x-3y=-23$

$$\begin{gathered} \Rightarrow 2x+3y=-13 and 2x+3y=23 \\ \Rightarrow 2x+3y+13=0 and 2x+3y-23. \end{gathered}$$