For Each Natural Number k, Let Ck denote The Circle With Radius k centimeters And Centre At The Origin O, On The Circle Ck a Particle Moves k centimeters In The Counter-clockwise Direction. After Completing Its Motion On Ck, The Particle Moves To Ck + 1 in The Radial Direction. The Motion Of The Particle Continues In This Manner. The Particle Starts At (1, 0). If The Particle Crosses The Positive Direction Of x-axis For The First Time On The Circle Cn’ then n =

Two points P and Q are taken on the line joining the points A (0, 0) and B (3a, 0) such that AP = PQ = QB. Circles are drawn onAP, PQ, and QB as diameters. The locus of the point S, the sum of the squares of the lengths of the tangents from which to the three circles is equal to b², is

If OA and OB are two equal chords of the circle x² + y² – 2x + 4y = 0 perpendicular to each other and passing through the origin O, the slopes of OA and OB are the roots of the equation 

An equation of the chord of the circle x² + y² = a² passing through the point (2, 3) farthest from the centre is

The lengths of the intercepts made by any circle on the coordinates axes are equal if the centre lies on the line (s) represented by

A circle touches both the coordinates axes and the line  the coordinates of the centre of the circle can be

If the tangent at the point P on the circle x² + y² + 6x + 6y = 2 meets the straight line 5x – 2y + 6 = 0 at a point Q on the y-axis, then the length ofPQ is

If a > 2b > 0 then the positive value of m for which  is a common tangent to x² + y² = b² and (x – a)²+ y² = b² is   

Let PQ  and RS be tangents at the extremities of a diameter PR of a circle of radius r. Such that PS and RQ intersect at a point X on the circumference of the circle, then diameter of the circle equals. 

A triangle PQR is inscribed in the circle x² + y² = 25. If Q and R have coordinates (3, 4) and (–4, 3) respectively, then ∠QPR is equal to

If the area of the quadrilateral formed by the tangent from the origin to the circle x² + y² + 6x – 10y + c = 0 and the pair of radii at the points of contact of these tangents to the circle is 8 square units, then c is a root of the equation