Question

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

Column I	Column II
(A) The equation of the locus of a point whose distance from the z-axis is equal to its distance from the xy-plane is	(p) x2 + y2 + z2 - 6x + 2y - 4z + 5 = 0
(B) The equation of the sphere with centre at (3, −1, 2) and touching yz-plane is	(q) y2 - 2y - 4x + 4z + 6 = 0
(C) The equation of the locus of the point whose distance from (2, −1, 3) is 4 is	(s) x2 + y2 - z2 = 0
(D) The equation of the locus of the point whose distance from the y-axis is equal to its distance from the point (2, 1, −1) is	(t) x2 + y2 + z2 - 4x + 2y - 6z - 2 = 0

Answer 1

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

Explanation :

(A) → (s)

(A) The distance of a point from z-axis is √(x² + y²). The distance of the point from xy-plane is |z|. Therefore x² + y² = z² or x² + y² − z² = 0

(B) Since the sphere touches yz-plane, its radius is |x|. Hence, the equation of the sphere is 

(x - 3)² + (y + 1)² + (z - 2)² = 3²

x² + y² + z² - 6x + 2y - 4z + 5 = 0

(C) The locus is

(x - 2)² + (y + 1)² + (z - 3)² = 16

x² + y² + z² -4x + 2y - 6z - 2 = 0

(D) We have

√(x² + z²) = √((x - 2)² + (y - 1)² + (z + 1)²)

x² + z² = (x - 2)² + (y - 1)² + (z + 1)²

y² - 4x - 2y + 2z + 6 = 0