Alg6.23

$A(2,3),B(-3,4)\,$ be given two points.Find the locus of P so that the area of the triangle PAB is 8.5square units.

Let P(x,y) be a point in locus.

The area of the triangle is

$\frac{1}{2}|[2(4-y)-3(y-3)+x(3-4)]|=\frac{17}{2}\,$

$|8-2y-3y+9-x|=17\,$

$|17-x-5y|=17\,$

$x+5y=0,x+5y=34\,$

Equation of the locus of P is

$(x+5y)(x+5y-34)=0\,$

$x^2+10xy+25y^2-34x-170y=0\,$