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…

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 Pα,β,γ, Q0,0,γ and Rα,β,-γ
Now, PQ→∥i^+j^ ⇒-αi^+βj^∥i^+j^
⇒α=β
Also, mid point of PQ lies on the plane
⇒α2+β2=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

Asked in: JEE Advanced 2018 (Paper 2)

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