Four Distinct Point (1, 0), (0, 1), (0, 0) And (t, t) Are Concyclic For

The centres of two circles C₁ and C₂ each of unit radius are at a distance of 6 units from each other. Let P be the mid-point of the line segment joining the centres of C₁ and C₂ and C be a circle touching C₁ and C₂externally. If a common tangent to C₁ and C passing through P is also a common tangent to C­₂ and C₁, then the radius of the circle C is 

An equation of the circle through (1, 1) and the points of intersection ofx² + y² + 13x – 3y = 0 and 2x² + 2y² + 4x – 7y – 25 = 0 is   

The abscissae of two points A and B are the roots of the equation x² + 2ax – b² = 0, and their ordinates are the roots of the equation x² + 2px –q² = 0. The radius of the circle with AB as diameter is  

The locus of the point of intersection of the tangent to the circle x = r cos θ, y = r sin θ at points whose parametric angles differ by

 is 

The locus of a point which moves such that the tangents from it to the two circles x² + y² – 5x – 3 = 0 and 3x² + 3y² + 2x + 4y – 6 = 0 are equal is 

If the two circles x² + y² + 2gx + 2fy = 0 and x² + y² + 2g₁x + 2f₁y = 0 touch each other, then

If two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 cut the coordinates axes in concyclic points, then 

The locus of the point which moves in a plane so that the sum of the squares of its distances from the lines ax + by + c = 0 and

bx – ay + d = 0 is r², is a circle of radius.  

The locus of the centre of the circle passing through the origin O and the points of intersection of any line through (a, b) and the coordinates axis is a

If two circles which pass through the points (0, a) and (0, –a) cut each other orthogonally and touch the straight line 

y = mx + c, then