Summary
This math recipe will help you find the equation of a line, whose following parameters are given:
- perpendicular distance of the line from the origin
- the angle made by the perpendicular to the line from the origin with the x-axis.
Skill Level
Easy
Time
Approx. 1 min
Ingredients
Perpendicular distance of the line from the origin: p
Angle made by the perpendicular (from the origin) with the x-axis: θ
Method
To find the equation of the line, we’ll use the following:
xcosθ + ysinθ = p
This is also known as the normal form of the equation of a line.
Examples
Example 1 Find the equation of a line whose perpendicular distance from the origin is √2, and the perpendicular to it from the origin makes a 45° angle with the x-axis.
Solution To find the equation, we’ll use the above form directly.
⇒ xcos45° + ysin45° = √2
⇒ x/√2 + y/√2 = √2
⇒ x + y = 2
The figure shows the line on the XY plane.
Example 2 Find the equation of a line whose perpendicular distance from the origin is 1, and the perpendicular to it from the origin makes a 120° angle with the x-axis.
Solution To find the equation, we’ll use the above form directly.
xcos120° + ysin120° = 1
⇒ -x/2 + y√3/2 = 1
⇒ -x + y√3 = 2
The figure shows the line on the XY plane.
Example 3 Find the equation of a line whose perpendicular distance from the origin is 3, and the perpendicular to it from the origin makes a 270° angle with the x-axis.
Solution To find the equation, we’ll use the above form directly.
⇒ xcos270° + ysin270° = 3
⇒ 0x – y = 3
⇒ y = -3
The figure shows the line on the XY plane.
That’s it for this recipe. Hope you found it helpful.
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