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.
My Self Assessment
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 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 In represents area of n sided regular polygon inscribed in a unit circle and On the area of n–sided regular polygon circumscribing it. Prove that
The line is a diameter of the circle
a) True
b) False
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 . . . . .
. 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.
