JEE Main 2025 Apr 2 Shift 2, Mathematics Q1: Lines in Space

Direction ratios of $AB$: $⟨3-4,5-7,3-1⟩=⟨-1,-2,2⟩$, or equivalently $⟨1,2,-2⟩$.

Write the line $AB$ in parametric form using point $B(3,5,3)$:

$(x,y,z)=(3+\lambda ,5+2\lambda ,3-2\lambda )$.

Let $R$ be the foot of perpendicular from $P(1,0,3)$ onto this line. Then $\overrightarrow{PR}\cdot ⟨1,2,-2⟩=0$:

$(\lambda +3-1)(1)+(2\lambda +5)(2)+(-2\lambda )(-2)=0$

$\Rightarrow \lambda +2+4\lambda +10+4\lambda =0\Rightarrow 9\lambda =-12\Rightarrow \lambda =-\frac{4}{3}$.

So $R=\left(\frac{5}{3},\frac{7}{3},\frac{17}{3}\right)$.

Since $R$ is the midpoint of $PQ$, $Q=2R-P=\left(\frac{7}{3},\frac{14}{3},\frac{25}{3}\right)$.

$\therefore \alpha +\beta +\gamma =\frac{7+14+25}{3}=\frac{46}{3}$.