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11. three dimension geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths11. three dimension geometry1 Mark
MCQ
The angle between the straight lines, whose direction cosines are given by the equations $2 l+2 \mathrm{~m}-\mathrm{n}=0$ and $\mathrm{mn}+\mathrm{n} l+l \mathrm{~m}=0$, is :
  • ✓
    $\frac{\pi}{2}$
  • B
    $\pi-\cos ^{-1}\left(\frac{4}{9}\right)$
  • C
    $\cos ^{-1}\left(\frac{8}{9}\right)$
  • D
    $\frac{\pi}{3}$
✓

Answer

Correct option: A.
$\frac{\pi}{2}$
a
$\mathrm{n}=2(\ell+\mathrm{m})$

$\ell \mathrm{m}+\mathrm{n}(\ell+\mathrm{m})=0$

$\ell \mathrm{m}+2(\ell+\mathrm{m})^{2}=0$

$2 \ell^{2}+2 \mathrm{~m}^{2}+5 \mathrm{~m} \ell=0$

$2\left(\frac{\ell}{\mathrm{m}}\right)^{2}+2+5\left(\frac{\ell}{\mathrm{m}}\right)=0$

$2 \mathrm{t}^{2}+5 \mathrm{t}+2=0$

$(\mathrm{t}+2)(2 \mathrm{t}+1)=0$

$\Rightarrow \mathrm{t}=-2 ;-\frac{1}{2}$

$(i)$ $\frac{\ell}{\mathrm{m}}=-2$

$\frac{\mathrm{n}}{\mathrm{m}}=-2$

$(-2 \mathrm{~m}, \mathrm{~m},-2 \mathrm{~m})$

$(-2,1,-2)$

$(ii)$ $\frac{\ell}{m}=-\frac{1}{2}$

$n=-2 \ell$

$(\ell,-2 \ell,-2 \ell)$

$(1,-2,-2)$

$\cos \theta=\frac{-2-2+4}{\sqrt{9} \sqrt{9}}=0 \Rightarrow 0=\frac{\pi}{2}$

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