JEE Main 2022 (Online) 30th July Evening ShiftQuestion 1
Let $OABC$ be a parallelogram, $O$ be the origin, $A(2,4,-5)$ and $C(b,2,3)$. If $P(a,a,a)$ and $Q(9-a^2,3,a-1)$, $a\in\mathbb{N}$, are two points such that the projection of $\overrightarrow{OP}$ on $\overrightarrow{OB}$ is $2$ and $\overrightarrow{OQ}$ makes acute angles with all three co-ordinate axes, then $\left|\overrightarrow{OB}\right|^2+\left|\overrightarrow{AC}\right|^2$ is equal to ________.
Vector Algebra
Answer$118$
Solution
$OABC$ parallelogram: $B=A+C=(2+b,6,-2)$. $\overrightarrow{OQ}$ makes acute angles with all axes: $9-a^2>0$, $a-1>0$, $a\in\mathbb N\Rightarrow a=2$, $P=(2,2,2)$. Projection $=\frac{2(2+b)+12-4}{|OB|}=\frac{2(b+6)}{|OB|}=2\Rightarrow(b+6)^2=(b+2)^2+36+4\Rightarrow8b=8\Rightarrow b=1$. $|OB|^2=9+36+4=49$; $\overrightarrow{AC}=(b-2,-2,8)=(-1,-2,8)$, $|AC|^2=69$. Sum $=118$.