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 locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

Let E be the ellipse  and C be the circle . Let P and Q be the points (1, 2) and (2, 1) respectively. Then

a) Q lies inside C but outside E

b) Q lies outside both C and E

c) P lies inside both C and E

d) P lies inside C but outside E

The radius of the circle passing through the focii of the ellipse  and having centre at (0, 3) is

a) 4

b) 3

c)

d)

Let P = (x, y) be any point on  with focii  and  equals

a) 8

b) 6

c) 10

d) 12

If tangents are drawn to the ellipse  then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

a)

b)

c)

d)

Multiple choice question

On the ellipse  the points at which the tangents are parallel to the line  are

a)

b)

c)

d)

Find the equation of the common tangent in the first quadrant to the circle  and the ellipse   . Also find the length of the intercept of the tangent between the coordinate axis.

Let P be a variable point on the ellipse  with foci F₁ and F_2. . If A is the area of  then the maximum value of A is  . . . . .

An ellipse has eccentricity  and one of the focus at the point  It’s one directrix is the common tangent near to the point P to the circle  and the hyperbola . Then the equation of the ellipse in the statement form is . . . . .

An ellipse has OB as a semi-minor axis. F, F’ are its focii and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is . . . . .

The equation  represents

a) An ellipse

b) A hyperbola

c) A circle

d) None of these

The equation  represents

a) No locus if k > 0

b) An ellipse if k < 0

c) A point if k = 0

d) A hyperbola if k > 0

If x = 9 is the chord of contact of the hyperbola  then the equation of the corresponding pair of tangents is

a)

b)

c)

d)

The curve described parametrically by   ,  represents

a) A pair of straight lines

b) An ellipse

c) A parabola

d) A hyperbola

For the hyperbola  which of the following remains constant with change in α

a) Abscissae of vertices

b) Abscissae of focii

c) Eccentricity

d) Directrix

If the line  touches the hyperbola  then the point of contact is

a)

b)

c)

d)

The circle  intersects hyperbola  in four points then

a)

b)

c)

d)