Question

A line is passing through (α, β, γ) and its direction cosines are l, m, n then the equations of the line are -

1. $\frac{x}{l}=\frac{\gamma }{m}=\frac{z}{n}$
2. $\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$
3. $\frac{x+\alpha }{l}=\frac{y+\beta }{m}=\frac{z+\gamma }{n}$
4. $\frac{x-\alpha }{l}=\frac{y+\beta }{m}=\frac{z-\gamma }{n}$

Answer: $\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$

Solution :-

If $l,m,n$ are the direction cosines of the line and that line passes through $\left(x_1,y_1,z_1\right)$ then the equation of the line in cartesian form is

$\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$ .

Now, In given question line is passing through $\left(\alpha ,\beta ,\gamma \right)$ and direction cosines are $l,m,n$ then the equations of the line are

$\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$ .

So, correct answer is option-2, $\to \frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$