Question

Let (p₁, q₁, r₁) and (p₂, q₂, r₂) are satisfying `|(1, p, p^2),(1, q, q^2),(1, r, r^2)|` = 6 (where p_i, q_i r_i ∈ N and p_i < q_i < r_i and i = 1, 2) and point (p₁, q₁, r₁) lies on the plane 2x + 3y + 6z = k, and point (p₂, q₂, r₂) lies on the plane 2x + 3y + 6z = k₂ (where p₁ = p₂ = 1) If distance between these planes is 'd', then value of (210d) is ______.

Options

• 60.00

• 70.00

• 80.00

• 90.00

Answer 1

Let (p₁, q₁, r₁) and (p₂, q₂, r₂) are satisfying `|(1, p, p^2),(1, q, q^2),(1, r, r^2)|` = 6 (where p_i, q_i r_i ∈ N and p_i < q_i < r_i and i = 1, 2) and point (p₁, q₁, r₁) lies on the plane 2x + 3y + 6z = k, and point (p₂, q₂, r₂) lies on the plane 2x + 3y + 6z = k₂ (where p₁ = p₂ = 1) If distance between these planes is 'd', then value of (210d) is 90.00.

Explanation:

(p – q)(q – r)(r – p) = 6

∴ p – q = –1,

q – r = –2,

r – p = 3

or p – q = –2,

q – r = –1,

r – p = 3

(p₁, q₁, r₁) = (1, 2, 4)

(p₂, q₂, r₂) = (1, 3, 4)

d = `|k_1 - k_2|/7 = 3/7`

210d = 90