Let O be any point in the plane of a circle, and OP₁P₂ any chord of the circle which passes through O and meets the circle in P₁ and P₂. On this chord is taken a point Q such that OQ is equal to the geometric mean between OP₁ and OP_2. Find the locus of Q.

Find the locus of the point of intersection of the tangent to a given circle and the perpendicular let fall on this tangent from a fixed point on a circle.

A straight line moves so that the product of the perpendiculars on it from the two fixed points is constant. Prove that the locus of the feet of perpendiculars from each of these points upon the straight line is a circle, the same for each.

O is a fixed point and AP, BQ are two fixed parallel straight lines; BOA is perpendicular to both and POQ is a right angle. Prove that the locus of the foot of perpendicular drawn from O upon PQ is a circle on AB as diameter.

Show that if the length of the tangents from a point P on the circle x² + y² = a² be four times the length of the tangent from it to the circle (x – a)² + y² = a², then P lies on the circle 15x² + 15y² – 32ax + a² = 0. Prove that these three circles pass through two points and that the distance between the centres of first and third circle is sixteen times the distance between the second and third circle.

In any circle prove that the square of the perpendiculars from any point on it to the line joining the point of contact of two tangents is proportional to the product of the perpendiculars from the point upon the two tangents.

Find the equations to the common tangents of the circles
x² + y² – 2x – 6y + 9 = 0 and x² + y² + 6x – 2y + 1 = 0

Find the equations to the common tangents of the circles x² + y² = c² and (x – a)² + y² = b²

Find the equation to the circle which passes through the origin and cuts orthogonally each of the circles
x² + y² – 6x + 8 = 0 and
x² + y² – 2x – 2y = 7

Prove that the square of tangents that can be drawn from any point on one circle to another circle is equal to twice the product of the perpendicular distance of the point from the radical axis of the two circles and the distance between their centres.

Find the general equation of all circles any pair of which have same radical axis as the circles x² + y².= 4 and x² + y² + 2x + 4y = 6.

Find the locus of the centre of the circles which cut the given two circles orthogonally.

Find the locus of a point which moves so that the length of the tangents drawn from it to one given circle is λ times length of tangent from it to another given circle.

If x + iy = tan (u + iv) where x, y, u, v are all real, prove that the curves u = constant gives a family of co–axial circles passing through the points (0, ±1)

If x + iy = tan (u + iv) where x, y, u, v are all real, prove that the curve v = constant gives a system of circles that cut orthogonally the system of co-axial circles represented by u = constant.

Find the equation to the circle which cuts orthogonally each of the circles
x² + y² + 2gx + c = 0
x² + y² + 2g'x + c = 0
and x² + y² + 2hx + 2ky + a = 0

Find the equation to the circle cutting orthogonally the circles
x² + y² – 2x + 3y – 7 = 0
x² + y² + 5x – 5y + 9 = 0
x² + y² + 7x – 9y + 29 = 0

Find the vertex, axis, focus and latus rectum of the parabola
4y² + 12x – 20y + 67 = 0

Find the vertex, axis, focus and latus rectum of the parabola
y² = 4y – 4x

Prove that the locus of the centre of a circle which intercepts a chord of given length 2a on the axis of x and passes through a given point on the axis of y distant b from the origin is the curve x² – 2by + b² = a²

If a circle be drawn so as always to touch a given straight line and also a given circle, prove that the locus of the centre is a parabola.

Vertex A of a parabola is joined to any point P on the curve and PQ is drawn at right angles to AP to meet the axis in Q. Prove that the projection of PQ on the axis is always equal to the latus rectum.

If on a given base, triangles be described such that the sum of the tangents of the base angles is constant, prove that the locus of the vertex is a parabola.