## Question

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.

### Solution

*x*^{2} + *y*^{2} + 32*x* + 4*y* + 235 = 0

The given circle and line are

*x*^{2} + *y*^{2} + 16*x* – 24*y* + 183 = 0 …(1)

and 4*x* + 7*y* + 13 = 0 …(2)

Centre and radius of circle (1) are (–8, 12) and 5 respectively. Let the centre of the imaged circle be (*x*_{1}, *y*_{1}).

Hence (*x*_{1}, *y*_{1}) be the image of the point (–8, 12) with respect to the line 4*x* + 7*y* + 13 = 0 then

∴ Equation of the imaged circle is

(*x* + 16)^{2} + (*y* + 2)^{2} = 5^{2}

or *x*^{2} + *y*^{2} + 32*x* + 4*y* + 235 = 0

#### SIMILAR QUESTIONS

Find the angle between the circles

Find the equation of the circle which cuts the circle *x*^{2} + *y*^{2} + 5*x* + 7*y* – 4 = 0 orthogonally, has its centre on the line *x* = 2 and passes through the point (4, –1).

Find the equations of the two circles which intersect the circles

*x*^{2} + *y*^{2} – 6*y* + 1 = 0 and *x*^{2} + *y*^{2} – 4*y* + 1 = 0

Orthogonally and touch the line 3*x* + 4*y* + 5 = 0.

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 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.