Different Forms of the Equation of Line

A line is a collection of all possible points that may be found on it. The relationship between the various points of the lines is shown in a line equation. A line equation specifies the requirements that must be met by the points. The line equation is an algebraic equation with two variables, x and y. A straight-line equation has x and y terms. If the equation of the line fulfills the point P(x,y), then the point P is on the line l.

Keywords: Straight Lines, Point of Origin, Slope, Coordinates, Collinearity, Intercepts, Plane, Perpendicular

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Different Forms of the Equations of Line

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1. Vertical Line

A vertical line at a distance of v from the y-axis has the equation x = v or x = -v. The location of the line determines the sign.

2. Horizontal Line

A horizontal line at a distance of h from the x-axis has the equation y = h or y = -h. The location of the line determines the sign.

3. Point-Slope Form

The requirement for a point to lie on the line is given by this version of the equation. Assume A (x0, y0) is a fixed point on the slope m line l. Consider another point, such as B. (x, y). If and only if, the coordinates of the point B (x, y) fulfill the equation, it will lie on the line l.

m = (y − y₀) / (x − x₀)

y − y₀ = m (x − x0)

4. Two-points Form

Consider a line l that connects the coordinates A (x₁, y₁) and B (x₂, y₂) (x₂, y₂). Consider the letter C. (x, y). If they fulfill the requirement of collinearity, this point will satisfy the condition to lie on.

AC Slope = BC Slope = AC Slope

Required equation= (y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁) or, (y − y₁) = (x − x₁) (y₂ − y₁) / (x₂ − x₁)

5. Slope-Intercept Form

The distance at which a line meets the y-axis is known as the y-intercept. An x-intercept is the point where a line intersects the x-axis. Assume a line l crosses the y-axis at a distance 'c' from the origin and has a slope of m. The equation for the line l that passes through the point (0, c) is.

m = (y − c) / (x − 0) or, y = mx + c

Consider a line l intersecting the x-axis at a point 'd' from the origin with a slope of m. The equation for the line l that passes through the point (d, 0) is.

m = (y − 0) / (x − d) or, y = m(x − d)

6. Intercept Form

Assume that a line l with slope m intersects the y-axis at a distance of 'c' from the origin and the x-axis at a distance of 'd'. The equation for the line l that passes through the points (0, c) and (d, 0) is

(y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁)

Or, (y − 0) / (x − d) = (c − 0) / (0 − d)

y/c + x/d = 1

7. Normal Form

Let's say we have a line l in the XY plane. The angle with the positive direction of the x-axis is known as well as the perpendicular distance (normal) from the origin to the line.

The length of the normal OA is p in this case, and it forms an angle with the x-axis. (p cosα, p sinα) will be the coordinates of A. The angle OA is perpendicular to the line l. tanα = slope of line l = -1 / slope of OA = -1 We get the needed line equation, x cosα + y sinα = p, after solving.

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General Equations of Line

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Ax + By + C = 0 is the universal equation for any line, where x and y are variables and A and B are not both zero. It's fascinating to learn that a line's general equation may be reduced to any type of equation.

1. Slope-Intercept Form

Ax + By + C = 0 may be simplified to y = (A/B) x C/B (of the form y = mx + c) if B ≠ 0. The y-intercept, c = C/B, and the slope of the line, m = -A/B. If B = 0, the x-intercept equals x = -C/A.

2. Intercept Form

Ax + By + C = 0 may be simplified to (x/ (-C/A)) + (y/ (-C/B)) = 1 (of the form x/a + y/b = 1) when C ≠ 0. The x-intercept is equal to -C/A, while the y-intercept is equal to -C/B. A = B = 0 if C = 0 and Ax + By = 0. It depicts a line that passes through the origin.

3. Normal Form

x cosα + y sinα = p is the standard form of the equation. We get cosα = A/(A² + B²)1/2, sinα = B/(A² + B²)1/2, and p = ±C /(A² + B²)1/2 when we reduce it to Ax + By + C = 0.

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Things to Remember

• A line in a coordinate plane forms two extra angles with the x-axis. The inclination of the line is the angle (θ) formed by the line l in the positive direction of the x-axis and measured anticlockwise.
• y = a, where a is the y – coordinate of the line's points, is the equation for horizontal or parallel lines to the X-axis. The equation x = a applies to a straight line that is vertical or parallel to the Y-axis, where a is the x-coordinate of the line's points.
• A general linear equation or general equation of a line is any equation of the form Ax + By + C = 0, where A and B are not both zero at the same time.
• The y axis is a straight line that passes through the origin and has a slope of 90°, 270°, or – 90°. So, depending on the sort of data we have for a line, straight lines may be represented in a variety of ways. The following are the several types of line of equations: Standard Form, General Form, Point-Slope Form, Slope-Intercept Form, and Two Point Form are the different types of forms.

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