Tangents TP and TQ are Drawn From A Point T to The Circle x2 + y2 = a2. If The Point T lies On The Line px + qy = r, Find The Locus Of Centre Of The Circumcircle Of Triangle TPQ.

Find the limiting points of the circles 

       (x² + y² + 2gx + c) + λ(x² + y² + 2fy + d) = 0

The circle x² + y² – 4x – 8y + 16 = 0 rolls up the tangent to it at  by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.  

Find the equation of the circle of minimum radius which contains the three circles 

                   x² – y² – 4y – 5 = 0 

               x² + y² + 12x + 4y + 31 = 0  

and         x² + y² + 6x + 12y + 36 = 0

Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.

If the line x cos α + y sin α = p cuts the circle x² + y² = a² in M and N, then show that the circle, whose diameter is MN, is 

A line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB. If the distance of the points A and B from the tangent at O, the origin, to the circle are m and n respectively, find the equation of the circle.

Tangents PQ, PR are drawn to the circle x² + y² = 36 from the point P(–8, 2) touching the circle at Q, R respectively. Find the equation of the circumcircle of the âˆ†PQR.

Let C­₁ and 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 centre of C.

The circle x² + y² – 4x – 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is x + y – xy + k(x² + y²)^1/2 = 0. Find k.

If the two lines  cut the co-ordinate axes in concyclic points, prove that a₁a₂ = b₁b₂ and find the equation of the circle.