QJEEM21Ma992a: JEE Main 2021 Physics - Class 11 JEEM21Ma992a Solution: Let O be the origin. Let and , x, y R, x > 0, be such that and…

$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k\,$

$\overrightarrow {OP} \bot \overrightarrow {OQ}$

$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$

$\overrightarrow {PQ} = \left( { - 1 - x} \right)\widehat i + \left( {2 - y} \right)\widehat j + \left( {3x + 1} \right)\widehat k$

$\left| {\overrightarrow {PQ} } \right| = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}}$

$\sqrt {20} = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}}$

20 = 1 + x² + 2x + 4 + y² $-$ 4y + 9x² + 1 + 6x

20 = 10x² + y² + 8x + 6 $-$ 4y

20 = 10x² + 4x² + 8x + 6 $-$ 8x

14 = 14x² $\Rightarrow$ x² = 1

Also, $\overrightarrow {OP} .\,\overrightarrow {OQ} = 0$

$- x + 2y - 3x = 0$

$4x = 2y$

y = 2x

$\therefore$ y² = 4x² $\Rightarrow$ y² = 4

x = 1 as x > 0 and y = 2

$\therefore$ $$\left|\begin{array}{ccc}x& y& -1\\ -1& 2& 3x\\ 3& z& -7\end{array}\right|=0$$

$\Rightarrow$ $$\left|\begin{array}{ccc}1& 2& -1\\ -1& 2& 3\\ 3& z& -7\end{array}\right|$$ = 0

$\Rightarrow$ 1( $-$ 14 $-$ 3z) $-$ 2(7 $-$ 9) $-$ 1( $-$ z $-$ 6) = 0

$\Rightarrow$ $-$ 14 $-$ 3z + 4 + z + 6 = 0

$\Rightarrow$ 2z = $-$ 4 $\Rightarrow$ z = $-$ 2

$\therefore$ x² + y² + z² = 9