If a point P is moving such that the lengths of tangents drawn from P to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ are in the ratio $2 : 3$ then find the equation of the locus of P.

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Solution

Equations of the circles are
$S = x^{2} + y^{2} - 4 x - 6 y - 12 = 0$
$S_{1} = x^{2} + y^{2} + 6 x + 18 y + 26 = 0$
$P (x_{1}, y_{1})$ is any point on the locus and $P T_{1}$ , $P T_{2}$ are the tangents from $P$ to the two circles.
Given condition is

$\frac{P T_{1}}{P T_{2}} = \frac{2}{3}$

$3 P T_{1} = 2 P T_{2}$

$3 (x^{2} + y^{2} - 4 x - 6 y - 12) = 2 (x^{2} + y^{2} + 6 x + 18 y + 26)$
$3 x^{2} + 3 y^{2} - 12 x - 18 y - 36 = 2 x^{2} + 2 y^{2} + 12 x + 36 y + 52)$
$x^{2} + y^{2} - 24 x - 54 y - 88 = 0$

This is the locus of $P (x_{1} - y_{1})$

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Q. If

P (x_{1}, y_{1})

is a point such that the lengths of the tangents from it to the circles

x^{2} + y^{2} - 4 x - 6 y - 12 = 0

and

x^{2} + y^{2} + 6 x + 18 y + 26 = 0

are in the ratio

2 : 3

then the locus of

P

The equation of the common tangent to the circles $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ at their point of contact.

Q. The number of common tangents to the circles

x^{2} + y^{2} - 4 x - 6 y - 12 = 0

and

x^{2} + y^{2} + 6 x + 18 y + 26 = 0

, is :

If the ratio of the lengths of tangents from a point to the circles $x^{2} + y^{2} + 4 x + 3$ = 0, $x^{2} + y^{2} - 6 x + 5$ = 0 Is 1:2 then the locus of P is a circle whose centre is

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If a point P is moving such that the lengths of tangents drawn from P to the circles x2+y2−4x−6y−12=0 and x2+y2+6x+18y+26=0 are in the ratio 2:3 then find the equation of the locus of P.

If a point P is moving such that the lengths of tangents drawn from P to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ are in the ratio $2 : 3$ then find the equation of the locus of P.