Find the equation of the circle which passes through the point (2, 0) and whose centre is the limit of the point of intersection of the lines .

Let A be the centre of the circle . Suppose the tangents at the points B (1, 7) and D (4, 2) on the circle meet at the point C, find the area of the quadrilateral ABCD.

Through a fixed point (h, k) secants are drawn to the circle  . Show that the locus of the mid points of the secant intercepted by the circle is

The abscissas of two points A and B are the roots of the equation  and their ordinates are the roots of the equation . Find the equation of the circle on AB as diameter.

Lines  and  touch a circle C₁ of diameter 6. If the centre of C₁ lies in the first quadrant, find the equation of the circle C₂ which is concentric with C₁ and cuts intercepts of length 8 on these lines.

Let a given line L₁ intersect the X-axis and Y-axis at P and Q respectively. Let another line L₂ perpendicular to L₁ cut the X and Y axis at R and S respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.

The circle  is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circum centre of the triangle is  find k.

Let  be a given circle. Find the locus of the foot of perpendicular drawn from the origin upon any chord of S which subtends a right angle at the origin.

If  are four points on a circle then show that .

The circles each of radius 5 units touch each other at (1, 2). If the equation of the common tangent is , find the equation of the circles.

Three circles touch each other externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4. Find the ratio of the product of the radii to the sum of the radii of the circles.

Let a circle be given by . Find the condition on a and b if two chords each bisected by the X–axis can be drawn from .

Let C be any circle with centre (0, . Prove that at the most two rational points can be there on C (A rational point is a point both of whose coordinates are rational numbers).

Consider a curve and a point P not on the curve. A line drawn from the point P intersects the curve at points Q and R. If PQ.QR is independent of the slope of the line then show that the curve is a circle.

Consider a family of circles . If in the first quadrant, the common tangent to a circle of the family and the ellipse  meet the coordinate axes at A and B, then find the locus of the mid-point of AB.

Let  be the equation of pair of tangents from the origin O to a circle of radius 3 with centre in the first quadrant. If A is a point of contact, find the length of OA.

Let C₁ , C₂ be two circles with C₂ lying inside C₁. A circle C lying inside C₁ touches C₁ internally and C₂ externally. Identify the locus of the center of C .

Find the point on   which is nearest to the line

Let I_n represents area of n sided regular polygon inscribed in a unit circle and O_n the area of n–sided regular polygon circumscribing it. Prove that

If A and B are points in the plane such that (constant) for all P on a given circle then the value of k cannot be equal to - -  - - -.

The points of intersection of the line  and the circle  is . . . . . 

The lines  and  are tangents to the same circle. The radius of this circle is . . . . .

From the origin chords are drawn to the circle . The equation of the locus of the mid points of these chords is . . . . .