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.

Two parabolas have the same axis and tangents are drawn to the second parabola from points on the first; prove that the locus of the middle points of chords of contact with the second parabola will all lie on a fixed parabola.

Prove that the locus of the foot of the perpendicular drawn from the vertex of the parabola upon chords which subtend an angle of 45° at the vertex, is the curve
r² – 24arcosθ + 16a²cos2θ = 0

Find the locus of a point which is such that two of the normals drawn from it to the parabola y² = 4ax are at right angles

If the normals to a parabola at three points P, Q and R meet in a point O and S be the focus, prove that
SP . SQ . SR = a . SO²

Three normals are drawn to the parabola y² = 4ax cosα from any point lying on the straight line y = b sinα. Prove that the locus of the orthocentre of the triangle formed by the corresponding tangents is the curve the angle α being variable.

Prove that the sum of the angles which the three normals to a parabola drawn from any point O, makes with the axis exceeds the angle which the focal distance of O makes with the axis by a multiple of π.

If the normals at the point P, Q and R meet in a point and if PP', QQ' and RR' be chords parallel to QR, RP and PQ respectively, prove that the normals at P', Q' and R' also meet in a point.

The normals at three points P, Q, R of the parabola y² = 4ax meet in a point O whose coordinates are (h, k). Prove that if P is fixed, then QR is fixed in direction and the locus of the centre of the circle circumscribing PQR is a straight line.

If the normals drawn from any point to the parabola y² = 4ax cut the line x = 2a in points whose ordinate are in arithmetic progression, prove that the tangents of the angles which the normals make with the axis are in geometric progression.

PG the normal at P to a parabola cuts the axis in G and is produced to Q so that GQ =  PG; prove that the other normals which pass through Q intersect at right angles.

Two parabolas y² = 4a (x – l) and x² = 4a (y – lʹ) always touch each other, the quantities l and lʹ being both variable; prove that the locus of their point of contact is the curve xy = 4a².

A parabola of latus rectum l, touches a fixed equal parabola y² = lx, the axes of the two parabolas being parallel, prove that the locus of the vertex of the moving curve is a parabola of latus rectum 2l.

Prove that the locus of the centre of the circle, which passes through the vertex of the parabola y² = 4ax and through the intersections with a normal chord, is the parabola 2y² = ax – a².

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.