Question

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).

Solution

Correct option is

x2 + y2 – 4x + 2y + 1 = 0

 

Let the required circle be

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

Since (4, –1) lie on (1) then

                 17 + 8g – 2f + c = 0                               …(2)

Centre of (1) is (–g, –f)

Since centre line on x = 2

Then                                 –g = 2 

∴                                         g = –2                          …(3)

From (2),               1 – 2f + c = 0                            …(4)

And given circle is 

                        x2 + y2 + 5x + 7y – 4 = 0                 …(5)

given the circles (1) and (5) cut each other orthogonally,  

  

  

                  

Solving (4) and (6), we get

                  = 1 and c = 1  

Substituting the values of gfc in (1), we get 

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

SIMILAR QUESTIONS

Q1

 

Find the equation of the diameter of the circle

x2 + y2 + 2gx + 2fy + c = 0 which corresponds to the chord ax = by + d= 0.

Q2

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.

Q3

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

Q4

 

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.

Q5

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.

Q6

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.

Q7

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.

Q8

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.

Q9

 

Find the angle between the circles

Q10

 

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.