Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment P Q is perpendicular to the plane x + y = 3 and the mid-point of P Q lies in the plane x + y = 3 ) lies on the z-axis. Let the distance of P from the x-axis be 5 . If R is the image of P in the xy-plane, then the length of P R is ______.

Step-by-step solution

Let P α , β , γ , Q 0,0 , γ and R α , β , - γ Now, P Q → ∥ i ^ + j ^   ⇒ - α i ^ + β j ^ ∥ i ^ + j ^ ⇒ α = β Also, mid point of PQ lies on the plane ⇒ α 2 + β 2 = 3 ⇒ α + β = 6 ⇒ α = 3 Now, distance of point P from x-axis is β 2 + γ 2 = 5 ⇒ β 2 + γ 2 = 25 ⇒ γ 2 = 16 As β = α = 3 As γ = 4 Hence, P R = 2 γ = 8