First slide

Straight lines in 3D

Question

Equation of the line $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 3}{4}$ is the line of intersection of -------

Moderate

A

$x + 2 y + 1 = 0 = 2 x - z + 1$
B

$x - y + 1 = 0 = x - 2 z + 4$
C

$2 x - y + 3 = 0 = x - y + z$
D

$x - y + z = 0 = x - y + 3 z$

Solution

The given line is $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 3}{4}$
Equate first two equations, $\frac{x - 1}{2} = \frac{y + 1}{- 1} \Rightarrow - x + 1 = 2 y + 2 \Rightarrow x + 2 y + 1 = 0$
Equate first and last equations $\frac{x - 1}{2} = \frac{z - 3}{4} \Rightarrow 2 (x - 1) = z - 3 \Rightarrow 2 x - z + 1 = 0$
Therefore, the given line is intersection of $x + 2 y + 1 = 0 = 2 x - z + 1$

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