Two parabolas have the same axis and tangents are drawn to the second parabola from points on the first; prove that the locus of the middle points of chords of contact with the second parabola will all lie on a fixed parabola.

Prove that the locus of the foot of the perpendicular drawn from the vertex of the parabola upon chords which subtend an angle of 45° at the vertex, is the curve
r² – 24arcosθ + 16a²cos2θ = 0

Find the locus of a point which is such that two of the normals drawn from it to the parabola y² = 4ax are at right angles

Find the locus of a point which is such that the three normals through it cut the axis in points whose distances from the vertex are in arithmetic progression.

If the normals to a parabola at three points P, Q and R meet in a point O and S be the focus, prove that
SP . SQ . SR = a . SO²

Three normals are drawn to the parabola y² = 4ax cosα from any point lying on the straight line y = b sinα. Prove that the locus of the orthocentre of the triangle formed by the corresponding tangents is the curve the angle α being variable.

Prove that the sum of the angles which the three normals to a parabola drawn from any point O, makes with the axis exceeds the angle which the focal distance of O makes with the axis by a multiple of π.

If the normals at the point P, Q and R meet in a point and if PP', QQ' and RR' be chords parallel to QR, RP and PQ respectively, prove that the normals at P', Q' and R' also meet in a point.

The normals at three points P, Q, R of the parabola y² = 4ax meet in a point O whose coordinates are (h, k). Prove that if P is fixed, then QR is fixed in direction and the locus of the centre of the circle circumscribing PQR is a straight line.

If the normals drawn from any point to the parabola y² = 4ax cut the line x = 2a in points whose ordinate are in arithmetic progression, prove that the tangents of the angles which the normals make with the axis are in geometric progression.

PG the normal at P to a parabola cuts the axis in G and is produced to Q so that GQ =  PG; prove that the other normals which pass through Q intersect at right angles.

Two parabolas y² = 4a (x – l) and x² = 4a (y – lʹ) always touch each other, the quantities l and lʹ being both variable; prove that the locus of their point of contact is the curve xy = 4a².

A parabola of latus rectum l, touches a fixed equal parabola y² = lx, the axes of the two parabolas being parallel, prove that the locus of the vertex of the moving curve is a parabola of latus rectum 2l.

Prove that the locus of the centre of the circle, which passes through the vertex of the parabola y² = 4ax and through the intersections with a normal chord, is the parabola 2y² = ax – a².

The area of the region bounded by the curve  and  is

a) 2

b) 3

c) 4

d) 6

Tangents OP and OQ are drawn from the origin O to the circle . Then the equation of the circumcircle of the ΔOPQ is

a)

b)

c)

d)

Paragraph question :  Based on the paragraph
Let A and  be two fixed points
V   and P moves on a plane such that .
If  then the locus of the point P is

a) A straight line

b) A circle

c) A parabola

d) An ellipse

Find the equation of the bisector of the obtuse angle between the lines  and

a)

b)

c)

d) None of these

The locus of middle point of the chords of the circle  

which subtend a right angle at the centre is

a)

b)

c)

d)

Let C be the centre of the hyperbola . The tangent at any point P on the hyperbola meets the straight line  and  at the points Q and R respectively. Then CQ.CR is equal to

a)

b)

c)

d)

The circle  is inscribed in a triangle which has two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is  then the value of k is equal to

a) 2

b) 1

c)

d)

The ends of a rod of length l move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio 1:2 is

a)

b)

c)

d) None of these

Find the equation to the ellipse whose centres are the origin, whose axes are axes of coordinates and which passes through the points (2, 2) and (3, 1).