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, \(\frac {23}{3}\)).