Question

If P₁ and P₂ are the lengths of the perpendiculars from the points (2, 3, 4) and (1, 1, 4) respectively from the plane 3x – 6y + 2z + 11 = 0, then P₁ and P₂ are the roots of the equation ______.

Options

• P² – 23P + 7 = 0

• 7P² – 23P + 16 = 0

• P² – 17P + 16 = 0

• P² – 16P + 7 = 0

Answer 1

If P₁ and P₂ are the lengths of the perpendiculars from the points (2, 3, 4) and (1, 1, 4) respectively from the plane 3x – 6y + 2z + 11 = 0, then P₁ and P₂ are the roots of the equation 7P² – 23P + 16 = 0.

Explanation:

P₁ = `|(3(2) - 6(3) + 2(4) + 11)/(sqrt(3^2 + (-6)^2 + (2)^2))|` = 1

P₂ = `|(3(1) - 6(1) + 2(4) + 11)/(sqrt(3^2 + (-6)^2 + (2)^2))| = 16/7`

The equation P₁ and P₂ satisfies 7P² – 23P + 16 = 0.

∴ P₁ and P₂ P₁ and P₂ are the roots of the equation (B).