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.

Let 2x² + y² – 3xy = 0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

If the circle C₁, x² + y² = 16 intersects another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to (3/4), find the co-ordinates of centre C₂.

The circle x² + y² = 1 cuts the x-axis at P and Q. another circle with centre at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of the triangleQSR.

Find the equation of a circle having the lines x² + 2xy + 3x + 6y = 0 as its normals and having size just sufficient to contain the circle

                         x(x – 4) + y(y – 3) = 0. 

Find the equation of the circle whose radius is 5 and which touches the circle 

              x² + y² – 2x – 4y – 20 = 0 at the point (5, 5).   

Find the locus of the mid point of the chord of the circle x² + y² = a²which subtend a right angle at the point (p, q).

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.

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.