Question

If A(-1,1) and B(2,3) are two fixed points, find the locus of a point P so that the area of $\Delta PAB$ =8sq.units.

Answer 1

$\text{Let P(h,k) be any point on the locus.}Then,Area\left(PAB\right)=8squnits\Rightarrow \frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|=8\Rightarrow \frac{1}{2}|-1(3-k)+2(k-1)+h(1-3)|=18\Rightarrow \frac{1}{2}|-3+k+2k-2-2h|=8\Rightarrow \frac{1}{2}|-2h+3k-5|=8\Rightarrow |-2h+3k-5|=16\Rightarrow -2h+3k-5=\pm 16\Rightarrow 2h+3k-5=\pm 16=0\Rightarrow 2h-3k+21=0or,2h-3k-11=0Hence,thelocusof(h,k)is2x-3y+21=0or,2x-3y-11=0$