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

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

Statement 1 – The line  is a diameter of the circle
Statement 2 – Normal of a circle always passes through the centre.
The question contains two statements: Statement 1 (assertion) and Statement 2 (reason), only one of which is the correct answer. You have to select one of the codes a, b, c, d given below.

a) Statement 1 is false, Statement 2 is true

b) Statement 1 is true, Statement 2 is true. Statement 2 is correct explanation of Statement1

c) Statement 1 is true, Statement 2 is true. Statement 2 is not a correct explanation of Statement1

d) Statement 1 is true, Statement 2 is false

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).