Question

 

Find the equation of the circle which cuts orthogonally each of the three circles given below: 

x2 + y2 – 2x + 3y – 7 = 0, x2 + y2 + 5x – 5y + 9 = 0 and x2 + y2 + 7x – 9x + 29 = 0.

Solution

Correct option is

x2 + y2 – 16x – 18y – 4 = 0

 

Let the required circle be  

             x2 + y2 + 2gx + 2fy + c = 0               …(1)

Since it is orthogonal to three given circles respectively, therefore

               

              

  

Subtracting (2) from (3),

               

and subtracting (3) from (4),

                2g – 4f = 20                                     …(6)

Solving (5) and (6), we get

                g = –8       and     f = –9

Putting the values of g and f in (3)

              –40 + 45 = c + 9

⇒                       5 = c + 9 

or                       c = – 4      

Substituting the values of gfc in (1) then required circle is

              x2 + y2 – 16x – 18y – 4 = 0

SIMILAR QUESTIONS

Q1

 

Find the equation of the system of circles coaxial with the circles.

              x2 + y2 + 4x + 2y + 1 = 0, x2 + y2 – 2x + 6y – 6 = 0

Also find the equation of that particular circle whose centre lies on radical axis.

Q2

Find the locus of pole of the line lx + my + n = 0 with respect to the circle which touches y-axis at the origin.

Q3

 

Find the circle whose diameter is the common chord of the circles

x2 + y2 + 2x + 3y + 1 = 0 and x2 + y2 + 4x + 3y + 2 = 0 

S ≡ x2 + y2 + 2x + 3y + 1 = 0 S’ ≡ x2 + y2 + 4x + 3y + 2 = 0

Q4

Find the equation of 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).

Q5

Find the point of intersection of the line 2x + 3y = 18 and the circle x2 +y2 = 25.

Q6

Find the equation of the normal to the circle x2 + y2 – 5x + 2y – 48 = 0 at point (5, 6).

Q7

Find the equation of the tangent to the circle x2 + y2 = 16 drawn from the point (1, 4).

Q8

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

Q9

Find the equation of the normal to the circle x2 + y2 = 2x, which is parallel to the line x + 2y = 3.

Q10

Circum centre of the triangle PT1T2 is at