Parabola

Determine the coordinates of the focus, the equation of the directrix and the length of the latus rectum of the parabola

y² = (8/3)x.

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

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

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

Passage: For the parabola y² = 4ax, the vertex is (0, 0), the focus is (a, 0) and the directrix is x + a = 0.

 Answer the following three questions.

(i) The vertex of the parabola (y – 1)² = 2(x – 1) is 

(A)  (1,0)

(B)  (2,0) 

(C)  (1,1)

(D)  (0,1)

(ii)  Focus of the parabola y² = 4(x – 1) is

(A)  (1, 0)

(B)  (2, 0)

(C)  (1, 1)

(D)  (2, 2)

(iii)  The directrix of the parabola (y – 2)² = 4(x – 1)

(A)   x = 1

(B)   x = -1

(C)   x = 2 

(D)  x =  0

If 2x + y + k = 0 is a normal to the parabola y² = − 8x, then the value of k is

(A)  12

(B)   24

(C)  − 12

(D)  − 24

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

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.