A Chord Of The Circle x2 + y2 – 4x – 6y = 0 Passing Through The Origin Subtends An Angle Tan-1 (7/4) At The Point Where The Circle Meets Positivey-axis. Equation Of The Chord Is

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

For each natural number k, let C_k denote the circle with radius k centimeters and centre at the origin O, on the circle C_k a particle moves k centimeters in the counter-clockwise direction. After completing its motion on C_k, the particle moves to C_{k + 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 C_n’ then n =

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

If two distinct chords, drawn from the point (p, q) on the circle x² + y²= px + qy (where pq ≠ 0) are bisected by the x-axis, then

Let A₀ A₁ A₃ A₄ A₅ be a regular hexagon inscribed in a unit circle with centre at the origin. Then the product of the lengths of the line segmentsA₀ A₁, A₀ A₂ and A₀ A₄ is

C₁ and C₂ are circles of unit radius with centres at (0, 0) and (1, 0) respectively. C₃ is a circle of unit radius, passes through the centres of the circles C₁ and C₂ and have its centre above x-axis. Equation of the common tangent to C₁ and C₃ which does not pass through C₂ is

On the line joining the points A (0, 4) and B (3, 0), a square ABCD is constructed on the side of the line away from the origin. Equation of the circle having centre at C and touching the axis of x is