The direction cosines $\ell, \mathrm{m}, \mathrm{n}$ of the line $\frac{\mathrm{x}+2}{2}=\frac{2…

The direction cosines $\ell, \mathrm{m}, \mathrm{n}$ of the line $\frac{\mathrm{x}+2}{2}=\frac{2 \mathrm{y}-4}{3} ; \mathrm{z}=-1$
  1. $\ell= \pm \frac{1}{\sqrt{5}}, \mathrm{~m}=0,= \pm \frac{2}{\sqrt{5}}$
  2. $\ell= \pm \frac{3}{5}, \mathrm{~m}= \pm \frac{4}{5}, \mathrm{n}=0$
  3. $\ell= \pm \frac{4}{5}, \mathrm{~m}= \pm \frac{3}{5}, \mathrm{n}=0$
  4. $\ell= \pm \frac{1}{\sqrt{3}}, \mathrm{~m}= \pm \frac{1}{\sqrt{3}}, \mathrm{n}= \pm \frac{1}{\sqrt{3}}$

Solution

We have line $\frac{x+2}{2}=\frac{2 y-5}{3}, z=-1$ i.e. $\frac{x-(-2)}{2}=\frac{2\left(y-\frac{5}{2}\right)}{3}, z=-1$ i.e. $\frac{x-(-2)}{2}=\frac{\left(y-\frac{5}{2}\right)}{\left(\frac{3}{2}\right)}, z=-1$ Here direction ratios are $2, \frac{3}{2}$ Also $\sqrt{(2)^2+\left(\frac{3}{2}\right)^2+0}= \pm \frac{5}{2}$ Here required direction cosines are $\frac{2}{\left( \pm \frac{5}{2}\right)}, \frac{\left(\frac{3}{2}\right)}{\left( \pm \frac{5}{2}\right)}, \frac{0}{\left( \pm \frac{5}{2}\right)} \text { i.e. } \pm \frac{4}{5}, \pm \frac{3}{5}, 0$

Asked in: MHT CET 2021 (21 Sep Shift 2)

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