The normal at the point P on the parabola y² = 4ax meets the axis in G and the tangent at the vertex in G'; if A is the vertex and the rectangle AGQG' is completed, prove that the equation to the locus of Q is x³ = 2ax² + ay².

Prove that the length of the chord joining the points of contact of tangents drawn from the point (, ) to the parabola y² = 4ax is

Prove that the area of the triangle formed by the tangents from the point (, ) to the parabola y² = 4ax and the chord of contact is   

If a perpendicular be let fall from any point P on a parabola upon its polar, prove that the distance of the feet of perpendicular from the focus is equal to the distance of the point P from the directrix.

If the tangents of a parabola at P and Q meet at T, prove that TP and TQ subtend equal angle at the focus.

Show that the area of the triangle formed by three points on a parabola is twice the area of triangle formed by the tangents at these points.

Prove that the orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.

If ω be the angle which a focal chord of a parabola makes with the axis, prove that the length of the chord is 4acosec²ω and that the perpendicular on it from the vertex is asinω

A point on a parabola, the foot of the perpendicular from it upon the directrix, and the focus are the vertices of an equilateral triangle. Prove that the focal distance of the point is equal to the latus rectum.

Prove that the semi-latus rectum is the harmonic mean between the segments of any focal chord.

If T be any point on the tangent at any point P of the parabola, and if TL be perpendicular to the focal radius SP and TN is perpendicular to the directrix, prove that SL = TN.

Prove that on the axis of any parabola there is a point K which has the property that, if a chord PQ of the parabola be drawn through it, then  is same for all positions of the chord.

The normal at the point  meets the parabola again in the point . Prove that  

A chord is a normal to the parabola and is inclined at an angle θ to the axis; prove that the area of triangle formed by it and the tangents at the extremities is 4a²sec³θcosec³θ.

If PQ be a normal chord of the parabola  and if S is the focus, prove that the locus of the centroid of the triangle SPQ is the curve  36ay² (3x – 5a) – 81y⁴ = 128a⁴.

Prove that the area of the triangle formed by the normals to the parabola  at the points
 is
 

Prove that the normal chord of a parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

A chord of the parabola passes through a point on the axis (outside the parabola) whose distance from the vertex is half the latus rectum; prove that the normals at the extremities meet on the curve.

The normal at a point P of the parabola meets the curve again in Q and T is the pole of PQ; show that T lies on the diameter passing through the other end of the focal chord passing through P and that PT is bisected by the directrix.

If from the vertex of the parabola  a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be made, prove that the locus of the farther angle of the rectangle is the parabola y² = 4a(x – 8a)

A series of chords of the parabola  is drawn so that their projections on a straight line which is inclined at an angle α to the axis are all of constant length c, prove that the locus of their middle point is the curve
(y² – 4ax) (y cosα + 2a sinα)² + a²c² = 0

Prove that the locus of the poles of chords of the parabola  which subtend a right angle at a fixed point (h, k) is
ax² – hy² + (4a² + 2ah)x – 2aky + a(h² + k²) = 0

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola  is
y² (2x + a) = a (3x + a)²