Find the length of the common chord of the circles, whose equations are
(x – a)² + y² = a²
x² + (y – b)² = b²

Prove that the equation to the circle whose diameter is this common chord is
(a² + b²) (x² + y²) = 2ab (bx + ay)

Prove that the following pair of circles intersect orthogonally
x² + y² – 2ax + c = 0 and x² + y² + 2by – c = 0

Find the radical axis of the pair of circles
x² + y² – 3x – 4y + 5 = 0 and
3x² + 3y² – 7x + 8y + 11 = 0

Find the radical axis of the set of circles
(x – 2)² + (y – 3)² = 36
(x + 3)² + (y + 2)² = 43
(x – 4)² + (y + 5)² =64

Find the equation to the straight lines joining the origin to the points of intersection of
x² + y² – 4x – 2y = 4 and
x² + y² – 2x – 4y – 4 = 0

Find the equation to the circle which cuts orthogonally each of the three circles
x² + y² + 2x + 17y + 4 = 0
x² + y² + 7x + 6y + 11 = 0
x² + y² – x +22y + 3 = 0

Find the equation to the circle cutting orthogonally the three circles
x² + y² = a²
(x – c)² + y² = a²
x² + (y – b)² = a²

Obtain the equation of the parabola whose focus is the point (2, 3) and whose directrix is the line x – 4y + 3 = 0

For what point of the parabola y² = 18x is the ordinate equal to three times the abscissa?

Prove that the equation to the parabola whose vertex and focus are on the axis of x at a distance a and a' from the origin respectively is y² = 4 (a' – a) (x – a)

In the parabola y² = 6x, find the equation to the chord through the vertex and the negative end of the latus rectum.

In the parabola y² = 6x, find the equation to any chord through the point on the curve whose abscissa is 24.

Prove that the locus of the middle point of all chords of parabola y² = 4ax which are drawn through the vertex is the parabola y² = 2ax.

PQ is a double ordinate of a parabola. Find the locus of its point of trisection.

A double ordinate of the curve y² = 4px is of length 8p; prove that the lines from the vertex to the two ends are at right angles.

Find the equation to the tangent to the parabola y² = 7x which is parallel to the straight line 4y – x + 3 = 0. Find also its point of contact.

A tangent to the parabola y² = 4ax makes an angle of 60° with the X – axis, find its point of contact.

Find points of the parabola y² = 4ax at which the tangent is inclined at 30° to the axis.

Find points of the parabola y² = 4ax at which the normal is inclined at 30° to the axis.

Find the equation to the tangents to the parabola y² = 9x, which goes through the point (4, 10).

Prove that x + y = 1 touches the parabola y = x – x².

Prove that the straight line y = mx + c touches the parabola y² = 4a (x + a) if  

Prove that the straight line lx + my + n = 0 touches the parabola y² = 4ax if ln = am²