Name: Perpendicular Distance Between Two Parallel Lines
Uploaded: 2019-07-19
Duration: 4 min 10 s
Description: The perpendicular distance between two lines is the perpendicular distance between any point on either one of them to the other line. Watch this video and le...

Question

Reduce the given equation into normal form. Find it's perpendicular distance from the origin and angle between the perpendicular and the positive

x

-axis.

x - y = 4

Accepted Answer

The Normal Form: The equation of a straight line upon which the length of the perpendicular from the origin is $p$ and angle made by this perpendicular to the $X$ -axis is $ω$ , is given by $x \cos ω + y \sin ω = p$ .
The given equation is $x - y = 4 .$
It can be reduced as $1 \cdot x + (- 1) y = 4$
On dividing both sides by $\sqrt{1^{2} + {(- 1)}^{2}} = \sqrt{2},$ we obtain
$\frac{1}{\sqrt{2}} x + (- \frac{1}{\sqrt{2}}) y = \frac{4}{\sqrt{2}} = 2 \sqrt{2}$ .......… $(i)$
On comparing equation $(i)$ with the normal form of equation of line
$x \cos ω + y \sin ω = p,$ we obtain $p = 2 \sqrt{2}$ , $\cos ω = \frac{1}{\sqrt{2}}$ , and $\sin ω = - \frac{1}{\sqrt{2}}$ . Since, $\cos$ is positive and $\sin$ is negative. $∴ ω$ lies in the $4^{t h}$ quadrant. So, $ω = 360^{o} - 45^{o} = 315^{o}$ .
So, the normal form of the given line is $x \cos 315^{o} + y \sin 315^{o} = 2 \sqrt{2} .$
Hence, the perpendicular distance of the line from the origin is $2 \sqrt{2},$ while the angle between the perpendicular and the positive $x$ -axis is $315^{o} .$

Reduce the given equation into normal form. Find it's perpendicular distance from the origin and angle between the perpendicular and the positive $x$ -axis.
$x - y = 4$

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Reduce the given equation into normal form. Find it's perpendicular distance from the origin and angle between the perpendicular and the positive x-axis. x-y=4

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Reduce the given equation into normal form. Find it's perpendicular distance from the origin and angle between the perpendicular and the positive $x$ -axis.
$x - y = 4$