Question

Match the items of Column I with those of Column II.

Column I	Column II
(A) The equation of the plane parallel to the plane passing through the points (1, 1, 1), (2, 3, 5) and (−1, 0, 2) which is at a distance of 2 units from the given plane is	(p) x + 2y + 3z = 3
(B) The equation of the plane passing through the intersection of the lines  and perpendicular to the line joining the point of their intersection and origin is	(q) 19(x + 5y - 4z) - 300 = 0
(C) A plane containing the line (x - 1)/1 = (y - 2)/2 = (z - 3)/3 is	(r) 2x - 3y + z + 2√14 = 0
(D) The equation of the plane which is parallel to the plane x + 5y − 4z + 5 = 0 and whose sum of the intercepts on the coordinate axes is 15 is	(s) 2x - 3y + z - 2√14 = 0
	(t) x + 2y + 3z - 14 = 0

Answer 1

(A) → (r),(s), (B) → (t), (C) → (p), (D) → (q)

Explanation :

(A) The equation of the plane passing through the points (1, 1, 1) (2, 3, 5) and (−1, 0, 2), from Theorem, is 

6(x + 1) - 9y + 3(z - 2) = 0

6x - 9y + 3z = 0

2x - 3y + z = 0

Any plane parallel to the above plane is of the form 2x − 3y + z + d = 0. Since the distance between those two planes is 2, we have

(B) Every point on the lines L₁ and L₂ is of the form P(1 + 3t, 2 + t, 3 + 2t) and Q(3 + s, 1 + 2s, 2 + 3s), respectively. Now P = Q gives

1 + 3t = 3 + s 3t - s = 2 ...(1)

2 + t = 1 + 2s  t - 2s = - 1 ...(2)

3 + 2t = 2 + 3s  2t - 3s = - 1 ...(3)

From the above equations t = 1 and s = 1 and the point of intersection of the lines is (4, 3, 5). Now vector OP(4,3,5). The required plane is

4(x - 4) + 3(y - 3) + 5(z - 5) = 0

4x + 3y + 5z - 50 = 0

(C) The line

(x - 1)/1 = (y - 2)/2 = (z - 3)/3

passes through (1, 2, 3). Since the plane passes through (1, 2, 3) which is having normal DRs (1, 2, 3), its equation is

1(x - 1) + 2(y - 2) + 3(z - 3) = 0

x + 2y + 3z - 14 = 0

(D) The plane parallel to the plane x + 5y − 4z + 5 = 0 is x + 5y − 4z = k. Now,

15 = Sum of the intercepts on the axes

= k + k/5 + k/-4

= (20k + 4k - 5k)/20 = (19k)/20

Therefore, k = 300/19. Hence, the equation of the plane

x + 5y - 4z = 300/19