Let co-ordinates of P be (x1,y1). Given that, x2+y2+2x−4y−20=0 . . . (i) and x2+y2−4x+2y−44=0 . . . (ii) Length of the tangent from P to Eq. (i) =x12+y12+2x1−4y1−20 . . . (iii) Length of the tangent from P to Eq. (ii) =x12+y12−4x1+2y1−44 . . . (iv) Given that ratio of lengths of tangent =
2
3
⇒
x12+y12+2x1−4y1−20
x12+y12−4x1+2y1−44
=
2
3
⇒3x12+3y12+6x1−12y1−60 =2x12+2y12−8x1+4y1−88 ⇒x12+y12+14x1−16y1+28=0 ∴ Locus of points P is x2+y2+14x−16y+28=0 Centre of the circle is (−7,8).