Question

Consider a triangle ABC whose vertices are A(0, α, α), B(α, 0, α) and C(α, α, 0), α > 0. Let D be a point moving on the line x + z – 3 = 0 = y and G be the centroid of ΔABC. If the minimum length of GD is `sqrt(57/2)`, then α is equal to ______.

Options

• 6

• 7

• 8

• 9

Answer 1

Consider a triangle ABC whose vertices are A(0, α, α), B(α, 0, α) and C(α, α, 0), α > 0. Let D be a point moving on the line x + z – 3 = 0 = y and G be the centroid of ΔABC. If the minimum length of GD is `sqrt(57/2)`, then α is equal to 6.

Explanation:

Centroid of ΔABC = `G((2α)/3, (2α)/3, (2α)/3)`

Given equation of line is `x/1 = (z - 3)/(-1) = y/0` = λ

x = λ, y = 0, z = – λ + 3

∴ D(λ, 0 – λ + 3) be any point on given line

∴ GD = `sqrt((λ - (2α)/3)^2 + ((2α)/3)^2 + (-λ + 3 - (2α)/3)^2`

GD₁ = `(λ - (2α)/3)^2 + ((2α)/3)^2 + (-λ + 3 - (2α)/3)^2`

`(dGD_1)/(dλ) = 2(λ - (2α)/3) - 2(-λ + 3 - (2α)/3)`

= 4λ – 6 = 0

`\implies` λ = `3/2`

∴ Minimum GD = `sqrt((3/2 - (2α)/3)^2 + ((2α)/3)^2 + (-3/2 + 3 - (2α)/3)^2`

`sqrt(57/2) = sqrt(((9 - 4α)/6)^2 + (4α^2)/9 + ((9 - 4α)/6)^2`

`\implies 57/2 = (24α^2 - 72α + 81)/18`

`\implies` α² – 3α – 18 = 0  ...(∵ α > 0)

`\implies` α = – 3, 6 

∴ α = 6