Geo4.32

From Example Problems
Jump to: navigation, search

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\,

Main Page:Geometry:Circles