Question

Find the equation of the line passing through the intersection of $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ and also through the point $(2,1,-2)$.

Answer 1

First line is,

$\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z-3}{4}=t$

$x=2t+1,y=-3t+2,z=4t+3$

Second line is,

$\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{1}=k$

$x=5k+4,y=2k+1,z=k$

For the point of intersection of the two lines,

$2t+1=5k+4$ ................... (1)

$-3t+2=2k+1$ ..................... (2)

$4t+3=k$ .....................(3)

Solving (1) and (2), we get,

$t=\frac{11}{19},$ $k=\frac{-7}{19}$

For the above values,

z-coordinate of the first line $=4t+3=\frac{101}{19}$

z-coordinate of the second line $=k=\frac{-7}{19}$

Which are not equal.

So, the lines are skew and do not intersect.

So, the equation of the line which passes through the intersection of these two given lines, cannot be found.