lf $p, q, r$ are the projections of $A B$ on the coordinate axes and $d_{1}, d_{2}, d_{3}$ are the projections of $A B$ on the coordinate planes then $d_{1}^{2} + d_{2}^{2} + d_{3}^{2} =$

2 p q r

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p^{2} + q^{2} + r^{2}

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2 (p^{2} + q^{2} + r^{2})

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\frac{p^{2} + q^{2} + r^{2}}{2}

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Solution

The correct option is C $2 (p^{2} + q^{2} + r^{2})$
Given $p, q, r$ are projections of $A B$ on coordinate axes.
$d_{1}, d_{2}, d_{3}$ are coordinate planes then $d_{1}^{2} + d_{2}^{2} + d_{3}^{2}$
Let $A$ be the origin and $B$ be the point in plane $(x, y, z)$
Formula: Projection of point $(x, y, z)$ on X-axis is $(x, 0, 0)$
On Y-axis is $(0, y, 0)$
On Z-axis is $(0, 0, z)$
Projection of point $(x, y, z)$ on XY plane is $(x, y, 0)$
On YZ plane is $(0, y, z)$
On ZX plane is $(x, 0, z)$
In this case
$p = x (4) q = y (5) z = r (6) d_{1} = \sqrt{x^{2} + y^{2}} (1) d_{2} = \sqrt{y^{2} + z^{2}} (2) d_{3} = \sqrt{x^{2} + z^{2}} (3)$
Squaring and adding $(1), (2), (3)$ equations
$d_{1}^{2} + d_{2}^{2} + d_{3}^{2}$ $= x^{2} + y^{2} + y^{2} + z^{2} + x^{2} + z^{2} = 2 (x^{2} + y^{2} + z^{2}) (1)$
From $(4), (5), (6)$
$d_{1}^{2} + d_{2}^{2} + d_{3}^{2}$ $= 2 (p^{2} + q^{2} + r^{2})$

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