## Question

If the origin be one limiting point of a system of co-axial circles of which*x*^{2} + *y*^{2} + 3*x* + 4*y* + 25 = 0 is a member, find the other limiting point.

### Solution

(–6, –8)

Equation of circle with origin as limiting point is

(*x* – 0)^{2} + (*y* – 0)^{2} = 0

or *x*^{2} + *y*^{2} = 0

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

*x*^{2} + *y*^{2} + 3*x* + 4*y* + 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

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

Find the equation of the circle through points of intersection of the circle*x*^{2} + *y*^{2} – 2*x* – 4*y* + 4 = 0 and the line *x* + 2*y* = 4 which touches the line *x*+ 2*y* = 0.

Find the circle whose diameter is the common chord of the circles *x*^{2} + *y*^{2}+ 2*x* + 3*y* + 1 = 0 and *x*^{2} + *y*^{2} + 4*x* + 3*y* + 2 = 0.

Find the angle between the circles

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

Find the equations of the two circles which intersect the circles

*x*^{2} + *y*^{2} – 6*y* + 1 = 0 and *x*^{2} + *y*^{2} – 4*y* + 1 = 0

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

Find the radical centre of circles *x*^{2} + *y*^{2} + 3*x* + 2*y* + 1 = 0,

*x*^{2} + *y*^{2} – *x* + 6*y* + 5 = 0 and *x*^{2} + *y*^{2} + 5*x* – 8*y* + 15 = 0. Also find the equation of the circle cutting them orthogonally.

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

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

*x*^{2} + *y*^{2} + 5*x* + *y* + 4 = 0 and *x*^{2} + *y*^{2} + 10*x* – 4*y* – 1 = 0

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