Question

Find the equation of a line which is perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and which cuts off an intercept of 4 units with the negative direction of y-axis.

Answer 1

The line perpendicular to \[\sqrt{3}x - y + 5 = 0\] is \[x + \sqrt{3}y + \lambda = 0\].

It is given that the line \[x + \sqrt{3}y + \lambda = 0\]  cuts off an intercept of 4 units with the negative direction of the y-axis.
This means that the line passes through \[\left( 0, - 4 \right)\].

\[\therefore 0 - \sqrt{3} \times 4 + \lambda = 0\]

\[ \Rightarrow \lambda = 4\sqrt{3}\]

Substituting the value of \[\lambda\], we get 

\[x + \sqrt{3}y + 4\sqrt{3} = 0\], which is the equation of the required line.