JEE Main 2026 — Three Dimensional Geometry Question with Solution

The direction vector of the line $L$ is given by the cross product of the two vectors to which it is perpendicular:

$\vec{d}=(2\hat{i}+2\hat{j}+\hat{k})\times (\hat{i}+2\hat{j}+2\hat{k})=2\hat{i}-3\hat{j}+2\hat{k}$

The equation of the line $L$ passing through $(1,1,1)$ is:

$\frac{x-1}{2}=\frac{y-1}{-3}=\frac{z-1}{2}=\lambda$

Any point $P$ on the line $L$ has coordinates $(2\lambda +1,-3\lambda +1,2\lambda +1)$.

Since $P(a,b,c)$ is the foot of the perpendicular from the origin $(0,0,0)$ to the line $L$, the vector $\overrightarrow{OP}$ is perpendicular to the direction vector $\vec{d}$ of the line. Thus, their dot product is zero:

$\overrightarrow{OP}\cdot \vec{d}=0$

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

$4\lambda +2+9\lambda -3+4\lambda +2=0$

$17\lambda +1=0\Rightarrow \lambda =-\frac{1}{17}$

Substituting $\lambda =-\frac{1}{17}$ into the coordinates of $P$, we get:

$a=2\left(-\frac{1}{17}\right)+1=\frac{15}{17}$

$b=-3\left(-\frac{1}{17}\right)+1=\frac{20}{17}$

$c=2\left(-\frac{1}{17}\right)+1=\frac{15}{17}$

The sum of the coordinates is:

$a+b+c=\frac{15}{17}+\frac{20}{17}+\frac{15}{17}=\frac{50}{17}$

Therefore, the value of $34(a+b+c)$ is:

$34\times \frac{50}{17}=100$

Answer: $100$