Find The Equation Of The Circle Passing Through (1, 0) And (0, 1) And Having The Smallest Possible Radius.

Let a circle be given by

                   2x (x – a) + y(2y – b) = 0            (a ≠ 0, b ≠ 0)

Find the condition on a and b if two chords, each bisected by the x-axis, can be drawn to the circle from (a, b/2).

The centre of the circle S = 0 lie on the line 2x – 2y + 9 = 0 and S = 0 cuts orthogonally the circle x² + y² = 4. Show that circle S = 0 passes through two fixed points and find their co-ordinates.

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

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

P is a variable on the line y = 4. Tangents are drawn to the circle x² + y²= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q.

Find the condition on a, b, c such that two chords of the circle

                x² + y² – 2ax – 2by + a² + b² – c² = 0  

passing through the point (a, b + c) are bisected by the line y = x.  

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

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