1
JEE Main 2024 (Online) 29th January Evening Shift
Numerical
+4
-1

Let O be the origin, and M and $$\mathrm{N}$$ be the points on the lines $$\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$$ and $$\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}$$ respectively such that $$\mathrm{MN}$$ is the shortest distance between the given lines. Then $$\overrightarrow{O M} \cdot \overrightarrow{O N}$$ is equal to _________.

2
JEE Main 2024 (Online) 29th January Morning Shift
Numerical
+4
-1

A line with direction ratios $$2,1,2$$ meets the lines $$x=y+2=z$$ and $$x+2=2 y=2 z$$ respectively at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. If the length of the perpendicular from the point $$(1,2,12)$$ to the line $$\mathrm{PQ}$$ is $$l$$, then $$l^2$$ is __________.

3
JEE Main 2024 (Online) 27th January Evening Shift
Numerical
+4
-1

The lines $$\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$$ and $$\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$$ intersect at the point $$P$$. If the distance of $$\mathrm{P}$$ from the line $$\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}$$ is $$l$$, then $$14 l^2$$ is equal to __________.

4
JEE Main 2023 (Online) 15th April Morning Shift
Numerical
+4
-1
Out of Syllabus
Let the plane $P$ contain the line $2 x+y-z-3=0=5 x-3 y+4 z+9$ and be

parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point

$\mathrm{A}(8,-1,-19)$ from the plane $\mathrm{P}$ measured parallel to the line $\frac{x}{-3}=\frac{y-5}{4}=\frac{2-z}{-12}$

is equal to ______________.