First slide

Direction ratios and direction cosines

Question

The direction cosines of a line satisfy the relations $λ (l + m) = n$ and $mn + nl + lm = 0 .$ the value of $λ$ for which the two lines are perpendicular to each other, is

Moderate

A

1
B

2
C

$\frac{1}{2}$
D

None of these

Solution

Eliminating n, we get
$\begin{array}{l} λ (l + m)^{2} + lm = 0 \\ \Rightarrow \frac{{λl}^{2}}{m^{2}} + (2 λ + 1) \frac{l}{m} + λ = 0 \\ \Rightarrow \frac{l_{1} l_{2}}{m_{1} m_{2}} = 1 (product of roots \frac{l_{1}}{m_{1}} and \frac{l_{2}}{m_{2}}) \end{array}$
where $\frac{l_{1}}{m_{1}}$ and $\frac{l_{2}}{m_{2}}$ are the roots of this equation, further eliminating m, we get
$\begin{aligned} {λl}^{2} - \ln - n^{2} = 0 \\ \Rightarrow \frac{l_{1} l_{2}}{n_{1} n_{2}} = - \frac{1}{λ} \end{aligned}$
Since the lines with direction cosines (l₁,m₁,n₁ and (1₂, m₂, n₂) are perpendicular,
we have $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$
$\begin{array}{l} or 1 + 1 - λ = 0 \\ or λ = 2 \end{array}$

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