A line makes an angle of $45^{\circ}$ with $x$ -axis and congruent angles with $y$ and $z$ -axes, then the…

A line makes an angle of $45^{\circ}$ with $x$ -axis and congruent angles with $y$ and $z$ -axes, then the direction cosines of the line are
  1. $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ and $-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$
  2. $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}$ and $-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}$
  3. $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$ and $-\frac{1}{\sqrt{2}},-\frac{1}{2},-\frac{1}{2}$
  4. $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$ and $\frac{1}{\sqrt{2}},-\frac{1}{2},-\frac{1}{2}$

Solution

Let angle made by line with each of $\mathrm{X}$ and $\mathrm{Z}$ axis be $\theta$. $\therefore \cos ^{2} 45^{\circ}+\cos ^{2} \theta+\cos ^{2} \theta=1$ $2 \cos ^{2} \theta=1-\frac{1}{2}=\frac{1}{2} \Rightarrow \cos ^{2} \theta=\frac{1}{4} \Rightarrow \cos \theta=\pm \frac{1}{2}$ Hence d.r.s. are $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$ or $\frac{1}{\sqrt{2}}, \frac{-1}{2}, \frac{-1}{2}$

Asked in: MHT CET 2020 (20 Oct Shift 2)

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