For a , b ∈ Z and | a - b | ≤ 10 , let the angle between the plane P : a x + y - z = b and the line L : x - 1 = a - y = z + 1 be cos - 1 1 3 If the distance of the point ( 6 , - 6 , 4 ) from the plane P is 3 6 , then a 4 + b 2 is equal to

Step-by-step solution

Given, For a , b ∈ Z and | a - b | ≤ 10 , let the angle between the plane P :   a   x + y - z = b and the line L :   x - 1 = a - y = z + 1 be cos - 1 1 3 Now rewriting the equation of line we get, L :   x - 1 1 = y - a - 1 = z + 1 1 Now let, θ = cos - 1 1 3 = sin - 1 2 2 3 Now we know that, Angle between plane and line is given by, cos 90 ° - θ = a + 1 - 1 + - 1 1 a 2 + 2 · 3 ⇒ a - 2 a 2 + 2 = 2 2 3 ⇒ 3 a - 2 2 = 8 a 2 + 2 ⇒ 3 a 2 + 4 - 4 a = 8 a