## Question

Find the lengths of common tangents of the circles *x*^{2} + *y*^{2} = 6*x* and *x*^{2} +*y*^{2} + 2*x* = 0. * *

### Solution

Equation of given circles are *x*^{2} + *y*^{2} = 6*x* and *x*^{2} + *y*^{2} + 2*x* = 0. The given equations of circles can be re-written as

(*x* – 3)^{2} + *y*^{2} = 9 and (*x* + 1)^{2} + *y*^{2} = 1

Centres and radii of the given circles are *C*_{1}(3, 0), *r*_{1} = 3 and

*C*_{2}(–1, 0), *r*_{2} = 1 respectively.

Circles touch to each other

Here internal tangent is impossible, only external tangent is possible.

#### SIMILAR QUESTIONS

Find the radical centre of circles *x*^{2} + *y*^{2} + 3*x* + 2*y* + 1 = 0,

*x*^{2} + *y*^{2} – *x* + 6*y* + 5 = 0 and *x*^{2} + *y*^{2} + 5*x* – 8*y* + 15 = 0. Also find the equation of the circle cutting them orthogonally.

Find the radical centre of three circles described on the three sides 4*x* – 7*y* + 10 = 0, *x* + *y* – 5 = 0 and 7*x* + 4*y* – 15 = 0 of a triangle as diameters.

Find the co-ordinates of the limiting points of the system of circles determined by the two circles

*x*^{2} + *y*^{2} + 5*x* + *y* + 4 = 0 and *x*^{2} + *y*^{2} + 10*x* – 4*y* – 1 = 0

If the origin be one limiting point of a system of co-axial circles of which*x*^{2} + *y*^{2} + 3*x* + 4*y* + 25 = 0 is a member, find the other limiting point.

Find the radical axis of co-axial system of circles whose limiting points are (–1, 2) and (2, 3).

Find the equation of the circle which passes through the origin and belongs to the co-axial of circles whose limiting points are (1, 2) and (4, 3).

Find the equation of the image of the circle *x*^{2} + *y*^{2} + 16*x* – 24*y* + 183 = 0 by the line mirror 4*x* + 7*y* + 13 = 0.

Find the area of the triangle formed by the tangents drawn from the point (4, 6) to the circle *x*^{2} + *y*^{2} = 25 and their chord of contact. Also find the length of chord of contact.

Find the lengths of external and internal common tangents to two circles*x*^{2} + *y*^{2} + 14*x* – 4*y* + 28 = 0 and *x*^{2} + *y*^{2} – 14*x* + 4*y* – 28 = 0.

Find the equation of the circle circumscribing the triangle formed by the lines:

*x* + *y* = 6, 2*x* + *y* = 4 and *x* + 2*y* = 5,

Without finding the vertices of the triangle.