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Question:

Let P be a point in the first octant, whose image Q in the plane x+y=3 (that is, the line segment PQ is perpendicular to the plane x+y=3 and the mid-point of PQ 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 PR is _____.

Solution:

Let P(α,β,γ), Q(0,0,γ) and R(α,β,-γ).
Now, PQ || ^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 √β² + γ² = 5 ⇒ β² + γ² = 25 ⇒ γ² = 16 as β = α = 3
as γ = 4
Hence, PR = 2γ = 8.