First slide

Direction ratios and direction cosines

Question

If the direction cosines of two lines are connected by the relations $l + 3 m + 5 n = 0 and 2 l^{2} - 5 m^{2} + n^{2} = 0$ , then the sum of all possible values of $\frac{m}{n}$ is

Difficult

A

$\frac{3}{13}$
B

$\frac{60}{13}$
C

$- \frac{30}{13}$
D

$- \frac{60}{13}$

Solution

The given relations are $l + 3 m + 5 n = 0 and 2 l^{2} - 5 m^{2} + n^{2} = 0$
From the first relation $l = (3 m + 5 n)$ , substitute this in the second equation
$\begin{matrix} 2 {(3 m + 5 n)}^{2} - 5 m^{2} + n^{2} = 0 \\ 18 m^{2} + 50 n^{2} - 5 m^{2} + n^{2} + 30 m n = 0 \\ 13 m^{2} + 51 n^{2} + 30 m n = 0 \end{matrix}$
Divide both sides by $n^{2}$
It gives
$13 {(\frac{m}{n})}^{2} + 30 (\frac{m}{n}) + 51 = 0$
Sum of all the values of $\frac{m}{n}$ is $- \frac{30}{13}$

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