Discussion
If α , β , γ are the angles made by the straight line \(\dfrac{x - 5}{2} = \dfrac{y + 1}{3} = \dfrac{z - 2}{\sqrt{7}}\) with the x-axis, y-axis and z-axis respectively, then \(\cos\alpha, \cos\beta, \cos\gamma\) are respectively,
Answer: E
The direction cosines are \(\dfrac{1}{\sqrt{5}}, \dfrac{3}{2\sqrt{5}}, \dfrac{\sqrt{7}}{2\sqrt{5}}\).
The denominators of the symmetric form give the direction ratios \(2, 3, \sqrt{7}\). Divide each by their length:
\[ \begin{aligned} \sqrt{4 + 9 + 7} &= \sqrt{20} = 2\sqrt{5} \\ \cos\alpha &= \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}} \end{aligned} \]
Similarly \(\cos\beta = \frac{3}{2\sqrt{5}}\) and \(\cos\gamma = \frac{\sqrt{7}}{2\sqrt{5}}\). Check: \(\frac{4 + 9 + 7}{20} = 1\). The options with a minus sign or with 7 or \(3\sqrt{7}\) in place of 3 or \(\sqrt{7}\) fail this check.
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