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   

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.

Find the equation to the common tangent of the circle x² + y² = 4ax and the parabola y² = 4ax.

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.

The value of k such that  lies in the plane
  is

a) 7

b) – 7

c) No real value

d) 4

If the lines  and  intersect then the value of k is

a)

b)

c)

d)

A variable plane at a distance of one unit from the origin cuts the coordinate axes at A, B and C. If the centroid D(x, y, z) of triangle ABC satisfies the relation  then the value of k is

a) 9

b)

c) 1

d) 3

A plane which is perpendicular to two planes  and  passes through (1, −2, 1). The distance of the plane from the point (1, 2, 2) is

a) 0

b) 1

c)

d)

Find the equation of the plane passing through the points (2, 1, 0), (4, 1, 1), (5, 0, 1). Find the point Q such that its distance from the plane is equal to the point P(2, 1, 6) from the plane and the line joining P and Q is perpendicular to the plane.

Two lines having direction ratios (1, 0, −1) and (1, −1, 0) are parallel to a plane passing through (1, 1, 1). This plane cuts the coordinate axes at A, B, C. Find the value of the tetrahedron OABC.