Let T be the pole of the chord PQ of a parabola; prove that the perpendicular from P, T, Q upon any tangent to the parabola are in geometric progression.

If r₁ and r₂ be the lengths of radii vectors of the parabola  which are drawn at right angles to one another from the vertex; prove that   

A circle is described on a focal chord of the parabola  as diameter, if m is the tangent of the inclination of the chord to the axis, prove that the equation of the circle is
 

LOLʹ and MOMʹ are two chords of a parabola passing through the point O on the axis. Prove that the radical axis of the circles described on LLʹ and MMʹ as diameter passes through the vertex of the parabola.

From the point where any normal to the parabola y² = 4ax meets the axis is drawn a line perpendicular to the normal. Prove that this line always touches an equal parabola.

Prove that the locus of the poles of chords that are normal to the parabola y² = 4ax is the curve
y² (x + 2a) + 4a³ = 0

Find the locus of the middle point of chords of the parabola y² = 4ax which subtend a right angle at the vertex, and prove that these chords will pass through a fixed point on the axis of the curve.

From an external point P tangents are drawn to the parabola y² = 4ax; find the equation of the locus of P when these tangents make angles θ₁ and θ₂ with the axis, such that
tanθ₁ + tanθ₂ is a constant (= b)

From an external point P tangents are drawn to the parabola y² = 4ax; find the equation of the locus of P when these tangents make angles θ₁ and θ₂ with the axis, such that
 θ₁ + θ₂ is a constant (= 2α)

From an external point P tangents are drawn to the parabola y² = 4ax; find the equation of the locus of P when these tangents make angles θ₁ and θ₂ with the axis, such that
cos θ₁ cosθ₂ is a constant (= μ)

Two tangents to the parabola y² = 4ax meet at an angle of 45°; prove that the locus of their point of intersection is the curve y² – 4ax = (x + a)². If they meet at an angle of 60°, prove that the locus is .

Prove that the locus of the point of intersection of two tangents to a parabola which intercept a given distance 4c on the tangent at the vertex is an equal parabola.

If the normals at P and Q meet on the parabola, prove that the point of intersection of the tangents at P and Q lie either on a certain straight line, which is parallel to the vertex or on the curve whose equation is y² (x + 2a) + 4a³ = 0

Two tangents to a parabola intercept on a fixed tangent segment whose product is constant; prove that the locus of their point of intersection is a straight line.

A point P is such that the straight line drawn through it perpendicular to its polar with respect to the parabola y² = 4ax touches the parabola x² = 4by. Prove that its locus is the straight line 2ax + by + 4a² =0

Two equal parabolas A and B have the same vertex and axis but their concavity turned in opposite directions; prove that the locus of poles with respect to B of tangent to A is the parabola A.

Prove that the locus of the pole of tangents to parabola y² = 4ax with respect to the circle x² + y² = 2ax is the circle x² + y² = ax.

Show that the locus of the poles of tangents to the parabola y² = 4ax with respect to the parabola y² = 4bx is the parabola

Find the locus of the middle points of the chord of the parabola y² = 4ax which passes through the focus.

Find the locus of the middle points of the chord of the parabola y² = 4ax which are normal to the curve.

Find the locus of the middle points of the chord of the parabola y² = 4ax which subtend a constant angle α at the vertex.

Find the locus of the middle points of the chord of the parabola y² = 4ax are of length l.

Find the locus of the middle points of the chord of the parabola y² = 4ax which are such that normals at their extremities meet on the parabola.