If the length of the perpendicular to a line from the origin is $2 \sqrt{2}$ units, which makes an angle of…
If the length of the perpendicular to a line from the origin is $2 \sqrt{2}$ units, which makes an angle of $135^{\circ}$ with the X -axis, then the equation of line is
$x+y=4$
$x-y+4=0$
$x-y=4$
$x+y+4=0$
Solution
Given that the perpendicular to the line makes an angle of $135^{\circ}$ with the X -axis.
$\therefore \quad$ Slope of perpendicular $=\tan \left(135^{\circ}\right)=1$
$\therefore \quad$ Slope of the required line $=1$, and it passes through the second quadrant.
$\therefore \quad$ Equation of the line is of the form $y=x+c$
i.e., $x-y+c=0$
Distance of this line from the origin is $2 \sqrt{2}$ units.
$\begin{array}{ll}
\therefore & 2 \sqrt{2}=\left|\frac{0-0+c}{\sqrt{(1)^2+(-1)^2}}\right| \\
\therefore & 2 \sqrt{2}=\left|\frac{c}{\sqrt{2}}\right|
\end{array}$
$\therefore \quad \mathrm{c}=4$ $\ldots[\mathrm{c}=-4$ is not possible as line passes through second quadrant]