Question

Find the coordinates of the point where the line through (5,1,6) and (3,4,1) crosses the ZX-plane.

Answer 1

It is known that the equation of the line passing through the points (x1,y1,z1) and (x2,y2,z2), is x-$\frac{x_1}{x_2}$-x1=y-$\frac{y_1}{y_2}$-y1=z-$\frac{z_1}{z_2}$-z1

The line passing through the points, (5,1,6) and (3,4,1), is given by,

$\frac{x-5}{3}$-5=$\frac{y-1}{4-1}$=$\frac{z-6}{1-6}$

⇒$\frac{x-5}{-2}$=$\frac{y-1}{3}$=$\frac{z-6}{-5}$=k(say)

⇒x=5-2k, y=3k+1, z=6-5k

Any point on the line is of the form (5-2k, 3k+1, 6-5k).

Since the line passes through ZX-plane.
3k+1=0

⇒k=$\frac{-1}{3}$

⇒5-2k

=5-2($\frac{-1}{3}$)

=$\frac{17}{3}$

⇒6-5k

=6-5($\frac{-1}{3}$)

=$\frac {23}{3}$

Therefore, the required point is ($\frac{17}{3}$ 0, $\frac {23}{3}$).