Two parabolas have  a common axis and concavity in opposite directions; if any line parallel to the common axis meet the parabolas in P and P', prove that the locus of the middle point on PP' is another parabola, provided that the latera recta of the given parabolas are unequal.

A parabola is drawn to pass through A and B, the ends of a diameter of a given circle of radius a, and to have as directrix a tangent to a concentric circle of radius b; the axes being AB and a perpendicular diameter, prove that the locus of the focus of the parabola is   

Write the equation to the tangent and normal at the end of the latus rectum of the parabola y² = 4a (x – a).

A tangent to the parabola y² = 8x makes an angle of 45° with the straight line y = 3x + 5. Find its equation and its point of contact.

Two equal parabolas have the same vertex and their axes are at right angles; prove that the common tangent touches at the end of the latus rectum.

Prove that the straight lines, one a tangent to the parabola y² = 4a (x + a) and other to the parabola y² = 4a' (x + a'), which are at right angles to one another, meet on the straight line x + a + a' = 0.

PN is an ordinate of the parabola; a straight line is drawn parallel to the axis to bisect NP and meet the curve in Q; prove that NQ meets the tangent at the vertex in a point T such that .

Let P, Q and R be three points of a parabola whose ordinates are in the geometric progression, prove that the tangent at P and R meet on the ordinate of Q.

Tangents are drawn to a parabola at points whose ordinates are in the ratio of  μ : 1. Prove they intersect on the curve

Find the length of normal drawn from a point on the parabola y² = 8ax whose distance from the focus is 8a.

Prove that the locus of the middle point of the portion of a normal intercepted between the curve and the axis is a parabola whose vertex is the focus and whose latus rectum is one quarter of that of the original parabola.

Prove that the distance between a tangent to the parabola y² = 4ax and parallel normal is cosecθ sec²θ, where θ is the angle that either makes with the axis.

PAP' is a double ordinate of the parabola y² = 4ax; prove that the locus of the point of intersection of the normals at P and the straight line through P' parallel to the axis is the equal parabola y² = 4a (x – 4a)

Prove that the parabola y² = 4ax and y² = 4c (x – b) cannot have a common normal, other than the axis, unless

Prove that the parabolas y² = 4ax and x² = 4by cut one another at an angle

Show that the two parabolas,
x² + 4a (y – 2b – a) = 0 and
y² = 4b (x – 2a + b)
intersect at right angles at the common end of the latus rectum of each.

Prove that three tangents to a parabola, which are such that the tangents of their inclinations to the axis are in a given harmonical progression, form a triangle whose area is constant.