Question

The distance of a variable plane from origin to plane is p and the Variable plane intersects the axis in A, B, C, then the point of intersection of given plane and the plane parallel to the co-ordinate plane is on (1 x²) + (1/y²) + (1/z²) = ____ 

(A) p²

(B) (1/p²) 

(C) p

(D) (1/p)   

Answer 1

The correct option  (B) (1/p²)  

Explanation:

eqⁿ of plane is xcosα + ycosβ + zcosγ = p.

it intersect axis A[{p/(cos α)}, 0, 0], B[0, {p/(cos β)}, 0] and C[0, 0, {p / (cos γ)}].

from A, B, C, equation of parallel plane 

x = {p/(cos α)}, y = {p/(cosβ)}, z = {p/(cos γ)} 

intersection of plane (x₁, y₁, z₁) = [{p/(cos α)}, {p/(cosβ)}, {p/(cos γ)}]

∴ cosα = (p/x₁), cosβ = (p/y₁), cos γ = (p/z₁)

α, β, γ is direction cosine.

cos²α + cos²β + cos²γ = 1

∴ p²[{1/(x₁²)} + {1/ (y₁²)} + {1/(z₁²)}] = 1

∴ {1/(x₁²)} + {1/(y₁²)} + {1/(z₁²)} = (1/p²).

∴ point of intersection of given plane and plane parallel to co-ordinate plane in on (1/x²) + (1/y²) + (1/z²) = (1/p²).