First slide

Direction ratios and direction cosines

Question

The equation of line passing through $(1, 5, 3)$ and having direction ratios $⟨ l, m, \frac{1}{2} ⟩$ in symmetric form is $\frac{x - 1}{l} = \frac{y - 5}{m} = \frac{z - 3}{\frac{1}{2}}$ then $l^{2} + m^{2} =$

Moderate

A

1
B

$\frac{1}{2}$
C

$\frac{3}{4}$
D

$\frac{1}{4}$

Solution

If $⟨ l, m, n ⟩$ are direction cosines of a line then $l^{2} + m^{2} + n^{2} = 1$
Hence, $l^{2} + m^{2} + {(\frac{1}{2})}^{2} = 1 \Rightarrow l^{2} + m^{2} = 1 - \frac{1}{4} = \frac{3}{4}$

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