Question

 

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.   

Solution

Correct option is

 

Let the required circle be

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

and given circles are     x2 + y2 – 6y +1 = 0                        …(2)

and                                x2 + y2 – 4y + 1 = 0                       …(3)

Since (1) cuts (2) and (3) orthogonally

   

Solving (4) and (5), we get

                     = 0 and c = –1  

Centre and radius of (6) are (–g, 0) and  respectively.

Since 3x + 4y + 5 = 0 is tangent of (6), then length of perpendicular from (–g, 0) to this line = radius of circle  

  

                                

   

  

Equations of circles are from (6),

             

              

          

SIMILAR QUESTIONS

Q1

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

Q2

Examine if the two circles x2 + y2 – 2x – 4y = 0 and x2 + y2 – 8y – 4 = 0 touch each other externally or internally.

Q3

 

Find the equation of the circle passing through (1, 1) and the points of intersection of the circles

x2 + y2 + 13x – 3y = 0 and 2x2 + 2y2 + 4x – 7y – 25 = 0.

Q4

Find the equation of the circle passing through the point of intersection of the circles x2 + y2 – 6x + 2y + 4 = 0, x2 + y2 + 2x – 4y – 6 = 0 and with its centre on the line y = x.

Q5

Find the equation of the circle passing through the points of intersection of the circles x2 + y2 – 2x – 4y – 4 = 0 and x2 + y2 – 10x – 12y + 40 = 0 and whose radius is 4.

Q6

Find the equation of the circle through points of intersection of the circlex2 + y2 – 2x – 4y + 4 = 0 and the line x + 2y = 4 which touches the line x+ 2y = 0.

Q7

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.

Q8

 

Find the angle between the circles

Q9

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).

Q10

 

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.