Geo4.32

Find the locus of the point from which the lenghts of the tangents to the circles $x^2+y^2+4x+3=0\,$ and $x^2+y^2-6x+5=0\,$ are in the ratio 2:3.

Given circles are

$x^2+y^2+4x+3=0,x^2+y^2-6x+5=0\,$

Let P(x1,y1) be a given point on locus.The lenghths of the tangents from P to the given circles are

$t_1=\sqrt{x_1^{2}+y_1^{2}+4x_1+3},t_2=\sqrt{x_1^{2}+y_1^{2}-6x_1+5}\,$

Given t1:t2=2:3, 3t1=2t2

$3t_1=2t_2, 9t_1^{2}=4t_2^{2}\,$

Hence

$9(x_1^{2}+y_1^{2}+4x_1+3)=4(x_1^{2}+y_1^{2}-6x_1+5)\,$

Simplifying

$a5x_1^{2}+5y_1^{2}+60x_1+7=0\,$

Hence the locus of P is the circle

$x^2+y^2+60x+7=0\,$