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.

If a chord which is normal to the parabola at one end subtends a right angle at the vertex, prove that it is inclined at an angle  to the axis.

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
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.