Question

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.

Solution

Correct option is

(–6, –8)

 

Equation of circle with origin as limiting point is  

                (x – 0)2 + (y – 0)2 = 0

or             x2 + y2 = 0

belongs to the system of co-axial circles of which one member is

                x2 + y2 + 3x + 4y + 25 = 0   

Hence the equation of the whole system is

               

Radius of (1) can be zero for limiting point, then

                 

             9 + 16 – 100(1 + λ) = 0    

  

or (–6, –8) is the other limiting point of the system.

SIMILAR QUESTIONS

Q1

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.

Q2

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.

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.

Q4

 

Find the angle between the circles

Q5

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).

Q6

 

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.   

Q7

 

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.

Q8

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.

Q9

 

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      

Q10

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