The axis of the parabola is along the line  and the distance of the vertex and focus from origin are  and  respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

a)

b)

c)

d)

(Multiple choice)

The equation of common tangent to the parabolas  and  is/are

a)

b)

c)

d)

Suppose that the normals drawn at three different points on the

parabola  pass through the point (h, 0). Show that h > 2.

A is a point on the parabola . The normal at A cuts the parabola again at B. If AB subtends a right angle at the vertex of the parabola, find the slope of AB.

Three normals are drawn from the point (c, 0) to the curve . Show that c must be greater than. One normal is always the X-axis. Find c for which the other two normals are perpendicular.

Through the vertex O of the parabola  chords OP and OQ are drawn at right angles. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the midpoint of PQ.

Show that the locus of a point that divides a chord of slope 2 of the parabola  internally in the ratio 1:2 is a parabola. Find its vertex.

Three normals with slopes  are drawn from a point P not on the axis of the parabola . If  results in the locus of P being a part of the parabola, find the value of α.

At any point P on the parabola  , a tangent is drawn which meets the directrix at Q. Find the locus of the point R which divides QP externally in the ratio  .

Match the following

Normals are drawn at the points P, Q and R lying on the parabola  which intersect at (3, 0) then

(Assertion and reason)

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

Statement 1 – The curve  is symmetric with respect to the line x = 1

Statement 2 – The parabola is symmetric about its axis.

The curve described parametrically by   ,  represents

a) A pair of straight lines

b) An ellipse

c) A parabola

d) A hyperbola

Obtain the equation of the parabola whose focus is the point (2, 3) and whose directrix is the line x – 4y + 3 = 0

For what point of the parabola y² = 18x is the ordinate equal to three times the abscissa?

Prove that the equation to the parabola whose vertex and focus are on the axis of x at a distance a and a' from the origin respectively is y² = 4 (a' – a) (x – a)

In the parabola y² = 6x, find the equation to the chord through the vertex and the negative end of the latus rectum.

In the parabola y² = 6x, find the equation to any chord through the point on the curve whose abscissa is 24.

Prove that the locus of the middle point of all chords of parabola y² = 4ax which are drawn through the vertex is the parabola y² = 2ax.

PQ is a double ordinate of a parabola. Find the locus of its point of trisection.

A double ordinate of the curve y² = 4px is of length 8p; prove that the lines from the vertex to the two ends are at right angles.

Find the equation to the tangent to the parabola y² = 7x which is parallel to the straight line 4y – x + 3 = 0. Find also its point of contact.

A tangent to the parabola y² = 4ax makes an angle of 60° with the X – axis, find its point of contact.

Find points of the parabola y² = 4ax at which the tangent is inclined at 30° to the axis.