Parabola

Equation of the normal at t is

tx + y = 2at + at³

If the line lx + my + n = 0 touches the parabola y² = 4ax, then

(a)   In = am

(b)  In² = am²

(c)  l²n = am²

(d)   In = am²

In general, three normals can be drawn through a point in the plane of a parabola to the curve.

Let L be the point (t, 2) and M be a point on the y-axis such that the slope of LM is −t. Then, the locus of the midpoint of LM is a parabola whose latus rectum is

(a)  2

(b)   1/2

(c)   4

(d)  1/4

Show that the locus of the midpoints of chords of a parabola passing through the vertex is in turn a parabola whose latus rectum is half of the latus rectum of the original. 

Show that the chords of contacts of points on the line 2x – 3y + 4 = 0 with respect to the parabola y² = 4ax pass through a fixed point.

A point P of the parabola y² = 4ax lies on the line y = x. The normal chord PQ, normal at P, subtends an angle at the focus of the parabola which equal

(A)  60º

(B)  45º

(C) 30º

(D)  90º

The foci of a hyperbola coincide with the foci of the ellipse \(\rm \frac{x^2}{25}+\frac{y^2}{16}=1\), find the equation of the hyperbola if its eccentricity is 3.

The orthocentre of a triangle formed by three tangents to a parabola y² = 8x lies on

(a)  x + 1 = 0

(b)  x + 2 = 0

(c)  x + = 0

(d)  x – 1 = 0