What is the equation to the chord of the parabola y² = 8x which is bisected at the point (2, −3).

The general equation to a system of parallel chords in the parabola  is 4x – y + k = 0. What is the equation to the

corresponding diameter?

If the tangents of a parabola at P and Q meet at T, prove that ST² = SP . SQ, where S is the focus.

If O is any point on the axis of a parabola and POP’ be any chord passing through O, and if PM and PʹMʹ be the ordinates of P and Pʹ, prove that AM . AMʹ =AO² and PM . PʹMʹ = −4a . AO, where A is the vertex.

Prove that all circles on focal radii of a parabola as diameter touch the tangent at the vertex.

A circle and parabola intersect in four points; show that the algebraic sum of the ordinates of the four points is zero.

Show that the line joining one pair of these points and the line joining the other pair are equally inclined to the axis.

Find the locus of the point O when the three normals to the parabola y² = 4ax from it are such that two of them make complementary angles with the axis.

Find the locus of the point O when the three normals to the parabola y² = 4ax from it are such that two of them make angles with the axis the product of whose tangents is 2.

Find the locus of the point O when the three normals to the parabola y² = 4ax from it are such that the sum of three angles made by them with the axis is constant (= λ).

The normals at three points P, Q and R of the parabola y² = 4ax meet in a point O whose coordinates are h and k, prove that the centroid of triangle PQR lies on the axis.

The normals at three points P, Q and R of the parabola y² = 4ax meet in a point O whose coordinates are h and k, prove that the sum of intercepts which the normals cut off from the axis is 2 (h + a).

A circle is described whose centre (0, 0) is the vertex and whose diameter is three quarters of the latus rectum of the parabola y² = 4ax. Prove that the common chord of the circle and parabola bisects the distance between the vertex and the focus.

Trace the curve
4x² – 4xy + y² – 12x + 6y + 9 = 0

A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is

a) (2, 0)

b) (0, 2)

c) (1, 0)

d) (0, 1)

Let P be the point (1, 0) and Q be any point on y² = 8x. Then the locus of the middle point of PQ is

a) x² – 4y + 2 = 0

b) x² + 4y + 2 = 0

c) y² + 4x + 2 = 0

d) y² – 4x + 2 = 0

If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the point of intersection of the parabolas y² = 4ax and x² = 4ay then

a) d² + (2b + 3c)² = 0

b) d² + (3b + 2c)² = 0

c) d² + (2b – 3a)² = 0

d) d² + (3b – 2a)² = 0

The normal at any point  on the parabola y² = 4bx meets the parabola again at the point , then

a)

b)

c)

d)

The portion of a tangent to a parabola y² = 4ax cut off between the directrix and the curve subtends an angle θ at the focus where θ is equal to

a)

b)

c)

d) None of these

The equation of the directrix of the parabola y² + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

The equation of the common tangent touching the circle
 and the parabola , above X–axis is

a)

b)

c)

d)

The equation of the common tangent to the curves  and  is

a)

b)

c)

d)

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

The angle between the tangents drawn from the point (1, 4) to the parabola  is

a)

b)

c)

d)