(A) If the line x - 1 = 0 is the directrix of the parabola y^2 - kx + 8 = 0, then the value of k

Question

 Match the items of Column I with those of Column II

Column I	Column II
(A)  If the line x - 1 = 0 is the directrix of the parabola y2 - kx + 8 = 0, then the value of k	(P)  2
(B) If l is the length of one side of an equilateral triangle inscribed in the parabola y2 = 4x with one vertex at the origin, then l/2√3 =	(q)  4
(C) The latus rectum of a parabola having (3, 5) and (3, −3) as extremities of the latus rectum	(r)  8
(D)  If (2, 0) is the vertex and y-axis as the directrix, then its focus is (a, 0) when a equals	(t)  -4
	(t)  -8

Answer 1

(A) The parabola y² - kx + 8 = 0 is written as

y² = k(x - 8/k)

= 4(k/4)(x - 8/k)

Hence, the directrix is

By hypothesis, x = 1 is the directrix. Therefore

Answer: (A) → (q), (t)

(B) From Problem 8 of the section ‘Subjective Problems’, we have l = 8a√3, where a = 1. Therefore

1/2√3 = 4

Answer: (B) → (q)

(C) The length of the latus rectum is

Answer: (C) → (r)

(D)  Distance of the vertex from the directrix (i.e., y-axis) is 2 and it is equal to half the distance of the focus from the directrix so that the focus is (4, 0). Therefore, a = 4.

Answer: (D) → (q)