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Relative Position of a Point with Respect to a Line

The direction...

Question

The direction cosines of the lines bisecting the angle between the lines whose direction cosines are $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ and the angle between these lines is $θ$ , are

A

\frac{l_{1} - l_{2}}{cos \frac{θ}{2}}, \frac{m_{1} - m_{2}}{cos \frac{θ}{2}}, \frac{n_{1} - n_{2}}{cos \frac{θ}{2}}

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B

\frac{l_{1} + l_{2}}{2 cos \frac{θ}{2}}, \frac{m_{1} + m_{2}}{2 cos \frac{θ}{2}}, \frac{n_{1} + n_{2}}{2 cos \frac{θ}{2}}

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C

\frac{l_{1} + l_{2}}{2 sin \frac{θ}{2}}, \frac{m_{1} + m_{2}}{2 sin \frac{θ}{2}}, \frac{n_{1} + n_{2}}{2 sin \frac{θ}{2}}

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D

\frac{l_{1} - l_{2}}{2 sin \frac{θ}{2}}, \frac{m_{1} - m_{2}}{2 sin \frac{θ}{2}}, \frac{n_{1} - n_{2}}{2 sin \frac{θ}{2}}

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Solution

The correct option is D $\frac{l_{1} - l_{2}}{2 sin \frac{θ}{2}}, \frac{m_{1} - m_{2}}{2 sin \frac{θ}{2}}, \frac{n_{1} - n_{2}}{2 sin \frac{θ}{2}}$
$cos θ = \frac{l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}}{\sqrt{l_{1}^{2} + m_{1}^{2} + n_{1}^{2}} \sqrt{m_{2}^{2} + n_{2}^{2} + l_{2}^{2}}}$
$\Rightarrow cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$
$\Rightarrow 2 cos^{2} \frac{θ}{2} - 1 = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$
$\Rightarrow cos \frac{θ}{2} = \sqrt{\frac{l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}}{2}}$
from figure, therefore the two angle bisector have direction ratios,
$l_{1} \pm l_{2}, m_{1} \pm m_{2}, n_{1} \pm n_{2}$
$∴$ direction cosines are: $\frac{1 {(l}_{1} \pm l_{2})^i + (m_{1} \pm m_{2})^j + (n_{1} \pm n_{2})^k}{\sqrt{{(l}_{1} \pm l_{2})^{2} + (m_{1} \pm m_{2})^{2} + (n_{1} \pm n_{2})^{2}}}$
$\Rightarrow 2 cos \frac{θ}{2} = \sqrt{2 (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})}$
$\Rightarrow 2 sin \frac{θ}{2} = \sqrt{2 (1 - l_{1} l_{2} - m_{1} m_{2} - n_{1} n_{2})}$
$\sqrt{{(l}_{1} \pm l_{2})^{2} + (m_{1} \pm m_{2})^{2} + (n_{1} \pm n_{2})^{2}} = 2 cos \frac{θ}{2}$
$\sqrt{{(l}_{1} \pm l_{2})^{2} + (m_{1} \pm m_{2})^{2} + (n_{1} \pm n_{2})^{2}} = 2 sin \frac{θ}{2}$
$∴$ The two directions cosines of two angles bisector are
$\frac{l_{1} + l_{2}}{2 cos \frac{θ}{2}}, \frac{m_{1} + m_{2}}{2 cos \frac{θ}{2}}, \frac{n_{1} + n_{2}}{2 cos \frac{θ}{2}}$ and
$\frac{l_{1} - l_{2}}{2 sin \frac{θ}{2}}, \frac{m_{1} - m_{2}}{2 sin \frac{θ}{2}}, \frac{n_{1} - n_{2}}{2 sin \frac{θ}{2}}$

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