Question

Find the equation of the image of the circle x2 + y2 + 16x – 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0.

Solution

Correct option is

x2 + y2 + 32x + 4y + 235 = 0

 

The given circle and line are

                    x2 + y2 + 16x – 24y + 183 = 0              …(1)

and              4x + 7y + 13 = 0                                  …(2)

Centre and radius of circle (1) are (–8, 12) and 5 respectively. Let the centre of the imaged circle be (x1y1).

Hence (x1y1) be the image of the point (–8, 12) with respect to the line 4x + 7y + 13 = 0 then

                   

∴ Equation of the imaged circle is  

                    (x + 16)2 + (y + 2)2 = 52  

or                 x2 + y2 + 32x + 4y + 235 = 0 

SIMILAR QUESTIONS

Q1

 

Find the angle between the circles

Q2

Find the equation of the circle which cuts the circle x2 + y2 + 5x + 7y – 4 = 0 orthogonally, has its centre on the line x = 2 and passes through the point (4, –1).

Q3

 

Find the equations of the two circles which intersect the circles

        x2 + y2 – 6y + 1 = 0 and x2 + y2 – 4y + 1 = 0  

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

Q4

 

Find the radical centre of circles x2 + y2 + 3x + 2y + 1 = 0,

x2 + y2 – x + 6y + 5 = 0 and x2 + y2 + 5x – 8y + 15 = 0. Also find the equation of the circle cutting them orthogonally.

Q5

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

Q6

 

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

  x2 + y2 + 5x + y + 4 = 0 and x2 + y2 + 10x – 4y – 1 = 0      

Q7

If the origin be one limiting point of a system of co-axial circles of whichx2 + y2 + 3x + 4y + 25 = 0 is a member, find the other limiting point.

Q8

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

Q9

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).

Q10

Find the area of the triangle formed by the tangents drawn from the point (4, 6) to the circle x2 + y2 = 25 and their chord of contact. Also find the length of chord of contact.