Find the locus of a point which moves so that the ratio of the lengths of the tangents to the circles `x^2+y^2+4x+3=0` and `x^2+y^2-6x+5=0` is `2: 3.`

Answer

`5x^(2)+5y^(2)+60x+7=0`

Explanation

Let the variable point be (h,k).
Accroding to the question,
`(sqrt(h^(2)+k(2)+4h+3))/(sqrt(h^(2)+k^(2)-6h+5))=(2)/(5)`
Squaring and simplifying, we get
`5h^(2)+5k^(2)+60h+7=0`
Hence, the required locus is `5x^(2)+5y^(2)+60x+7=0`