First slide

Straight lines in 3D

Question

The equation of the passing through $(1, 0, 3)$ and having direction cosines $a, b, \frac{1}{2}$ in
Symmetric form is $\frac{x - 1}{a} = \frac{y}{b} = \frac{z - 3}{\frac{1}{2}}$ then $a^{2} + b^{2}$ is

Easy

A

$\frac{3}{4}$
B

$\frac{1}{2}$
C

$\frac{1}{4}$
D

$0$

Solution

The Equation of the line passing through the point $(x_{1}, y_{1}, z_{1})$ and having direction ratios $⟨ a, b, c ⟩$ is $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$
The equation of the line passing through the point $(1, 0, 3)$ and having direction cosines $a, b, \frac{1}{2}$ is $\frac{x - 1}{a} = \frac{y - 0}{b} = \frac{z - 3}{\frac{1}{2}}$
since $a, b, \frac{1}{2}$ are direction cosines of the line, then $a^{2} + b^{2} + \frac{1}{4} = 1 \Rightarrow a^{2} + b^{2} = \frac{3}{4}$

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