If the direction cosines of a line are left(f | vedclass

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(B) The sum of the squares of the direction cosines of a line is always $1$. Given direction cosines are $\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \f

Vedclass · Accepted Answer

$21$

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(B) The sum of the squares of the direction cosines of a line is always $1$. Given direction cosines are $\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}$. Therefore,$\left(\frac{a}{\sqrt{83}}\right)^2 + \left(\frac{5}{\sqrt{83}}\right)^2 + \left(\frac{c}{\sqrt{83}}\right)^2 = 1$. $\Rightarrow \frac{a^2}{83} + \frac{25}{83} + \frac{c^2}{83} = 1$. $\Rightarrow a^2 + 25 + c^2 = 83$. $\Rightarrow a^2 + c^2 = 58$ ...$(i)$. Given $c - a = 4$,squaring both sides gives $(c - a)^2 = 16$. $c^2 + a^2 - 2ca = 16$. Substituting $a^2 + c^2 = 58$ from equation $(i)$: $58 - 2ca = 16$. $2ca = 58 - 16 = 42$. $ca = 21$.

If the direction cosines of a line are $\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$ and $c-a=4$,then $ca=$

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