Alg6.25

Find the locus of P,if the ratio of the distances from P to $(5,-4),(7,6)\,$ is 2:3.

Let P be (x,y) and the given points be A,B.The given geometric condition is

$\frac{PA}{PB}=\frac{2}{3}\,$

$3PA=2PB\,$

Squaring on both sides,

$9PA^{2}=4PB^{2}\,$

$(x-5)^2+(y+4)^2=(x-7)^2+(y-6)^2\,$

$x^2+25-10x+y^2+16+8y=x^2+49-14x+y^2+36-12y\,$

$14x-10x+8y+12y=85-25-16\,$

$4x+20y=44\,$

$x+5y=11\,$