the length of the perpendicular from the origin to a line is `7` and a line makes an angle of `150^@` with the positive direction of `y`-axis . then the equation of the line is:

Explanation

The equation of line in the normal form is `x\cos \alpha+y\sin \alpha=p`

Here, `\alpha=(150^\circ+90^\circ)` as angle is given with positive `y`-axis.

`\implies \alpha=240^\circ` and `p=7`

Therefore, required equation of line is,
`x\cos 240^\circ+y\sin 240^\circ=7`

`\implies x\frac{-1}{2}+y\frac{-\sqrt3}{2}=7`

`\implies -x-\sqrt3y=14`

`\implies x+\sqrt3y+14=0`