Straight Line — Forms

Stand on a straight road that climbs a hill. For every 100 metres you walk forward (horizontally), the road rises 15 metres. The steepness of that road is 15/100 = 0.15. Walk 200 metres forward, the road rises 30 metres — the ratio stays the same.

That ratio — rise divided by run — is called the slope of the road. It is a single number that completely describes the steepness and direction of any straight path. A flat road has slope 0. A road that goes uphill to the right has a positive slope. A road going downhill to the right has a negative slope. A steeper hill has a larger slope.

Slope is the first thing you need to talk about straight lines in coordinate geometry. Once you have it, every equation of a straight line is just slope plus one more piece of information — a point the line passes through, or a distance from the origin, or where it crosses the axes.

Slope of a line

The definition

Take two points on a line: A(x_1, y_1) and B(x_2, y_2). The slope of the line through A and B is:

The numerator y_2 - y_1 is the vertical change — how much the line rises (or falls) between the two points. The denominator x_2 - x_1 is the horizontal change — how far you move along the x-axis. The ratio is the rise per unit of run.

A key fact: the slope does not depend on which two points you pick. If you pick different points C and D on the same line, you get the same slope. This is because all the right triangles you can form between points on a line are similar — same angles, same ratios. Slope is a property of the line, not of the pair of points.

Slope and angle of inclination

The angle of inclination of a line is the angle \theta that the line makes with the positive x-axis, measured counterclockwise. The slope and the angle are related by:

This makes sense geometrically. In the right triangle formed by the rise and run, the angle at the base is \theta. The tangent of that angle is opposite/adjacent = rise/run = m.

Some important cases:

• A horizontal line has \theta = 0° and m = \tan 0° = 0.
• A line at 45° to the x-axis has m = \tan 45° = 1.
• A line at 135° (sloping downward) has m = \tan 135° = -1.
• A vertical line has \theta = 90° and m = \tan 90°, which is undefined. A vertical line has no slope — it is infinitely steep. This is why the slope formula has x_2 - x_1 in the denominator: when the line is vertical, x_2 = x_1 and you divide by zero.

Parallel and perpendicular lines

Two lines are parallel if and only if they have the same slope: m_1 = m_2.

Two lines are perpendicular if and only if the product of their slopes is -1: m_1 \cdot m_2 = -1, or equivalently m_2 = -1/m_1.

The perpendicularity condition deserves a quick proof. If line 1 makes angle \theta_1 with the x-axis and line 2 makes angle \theta_2 = \theta_1 + 90°, then:

(using \tan(\theta + 90°) = -\cot\theta). The condition m_1 m_2 = -1 follows directly.

The six forms of a straight line

1. Slope-intercept form

If you know the slope m and the y-intercept c (the value of y where the line crosses the y-axis), the equation is:

Why this works. The line crosses the y-axis at (0, c). From there, for every unit you move right, y increases by m. So at a general point x, the value of y is c + m \cdot x.

Example. A line with slope 2 and y-intercept -3: y = 2x - 3.

This is the most commonly used form because it is the simplest: just two parameters, both with clear geometric meanings. It is also the form you use when graphing — plot the y-intercept, then use the slope to find a second point.

Limitation. This form cannot represent a vertical line (x = k), because a vertical line has undefined slope. Every other line in the plane has a unique slope-intercept representation.

2. Point-slope form

If you know the slope m and one point (x_1, y_1) on the line, the equation is:

Derivation. Take any other point (x, y) on the line. The slope between (x_1, y_1) and (x, y) must equal m:

Multiply both sides by (x - x_1):

That is the entire derivation — one application of the slope definition. This form is the most useful when you are given a point and a slope (or can compute the slope from some other condition, like parallelism or perpendicularity).

3. Two-point form

If you know two points (x_1, y_1) and (x_2, y_2) on the line, first compute the slope:

Then use the point-slope form with either point:

This is not a separate formula — it is the point-slope form with the slope written out explicitly. Some textbooks present it as a separate "form," but it is really just the obvious combination of the slope formula and the point-slope equation.

4. Intercept form

If you know the x-intercept a (where the line crosses the x-axis) and the y-intercept b (where the line crosses the y-axis), the equation is:

Derivation. The line passes through (a, 0) and (0, b). Compute the slope: m = \frac{b - 0}{0 - a} = -\frac{b}{a}.

Use the point-slope form with the point (a, 0):

Divide both sides by b:

Verification. At (a, 0): \frac{a}{a} + \frac{0}{b} = 1. At (0, b): \frac{0}{a} + \frac{b}{b} = 1. Both check out.

This form is compact and elegant. It is especially useful in problems where the intercepts are the given data — for instance, "a line cuts the positive x-axis at 3 and the positive y-axis at 4."

Limitation. This form cannot represent lines through the origin (both intercepts are zero), or lines parallel to an axis (one intercept is infinite).

5. Normal form

This is the most elegant form, and the least intuitive on first encounter. Instead of describing the line by its slope and a point, you describe it by its distance from the origin and the direction of the perpendicular from the origin to the line.

Let p be the perpendicular distance from the origin to the line, and let \alpha be the angle that this perpendicular makes with the positive x-axis. Then the equation of the line is:

where p > 0.

Derivation. The foot of the perpendicular from the origin to the line is the point N(p\cos\alpha, p\sin\alpha) — this is the point at distance p from the origin in the direction \alpha.

The line is perpendicular to ON at N. The direction of ON is (\cos\alpha, \sin\alpha), so the line is perpendicular to this direction.

A point (x, y) lies on the line if and only if the vector from N to (x, y) is perpendicular to the direction (\cos\alpha, \sin\alpha). The perpendicularity condition (dot product equals zero) is:

So x\cos\alpha + y\sin\alpha = p.

The normal form may seem like an odd way to describe a line, but it has a major advantage: the distance p appears explicitly in the equation. This makes it trivially easy to compute the distance from the origin to any line — just convert to normal form and read off p. It is also the form used in many physics and engineering contexts, where the "closest approach" distance is what matters.

Converting between forms

Any line can be written in any of these forms (with the noted exceptions for vertical lines and lines through the origin). Here is a summary of how to convert:

Slope-intercept to normal form. Start with y = mx + c, rewrite as mx - y + c = 0. The normal form requires dividing by \sqrt{m^2 + 1} (the magnitude of the coefficient vector (m, -1)) and choosing the sign so that p > 0:

Here \cos\alpha = \frac{m}{\sqrt{m^2+1}} and \sin\alpha = \frac{-1}{\sqrt{m^2+1}}, with the sign of the whole expression chosen so that p = \frac{|c|}{\sqrt{m^2+1}} > 0.

Intercept form to slope-intercept. From x/a + y/b = 1: solve for y to get y = -\frac{b}{a}x + b. So the slope is -b/a and the y-intercept is b.

Normal form to slope-intercept. From x\cos\alpha + y\sin\alpha = p: solve for y to get y = -\frac{\cos\alpha}{\sin\alpha}x + \frac{p}{\sin\alpha}. So the slope is -\cot\alpha and the y-intercept is p/\sin\alpha.

Worked examples

How to choose the right form

Here is a decision guide. Given a problem about a straight line, look at what information you have, and pick the form that uses that information directly.

The forms are not competing — they are the same object viewed from different angles. The slope-intercept form is the one you will use most, because slope and y-intercept are the most commonly available data. The normal form is the most elegant but the least commonly needed at the introductory level.

Special lines

Some lines deserve special mention because they come up constantly.

Horizontal lines. y = k for some constant k. Slope is 0. Parallel to the x-axis.

Vertical lines. x = k for some constant k. Slope is undefined. Parallel to the y-axis. Cannot be written in slope-intercept form.

Lines through the origin. y = mx — the slope-intercept form with c = 0. These cannot be written in intercept form because both intercepts are 0 (and you cannot divide by zero).

The line y = x. Slope 1, angle of inclination 45°. This line is the locus of points with equal x- and y-coordinates.

The line y = -x. Slope -1, angle of inclination 135°. Perpendicular to y = x.

Common confusions

• "The slope of a vertical line is zero." No — the slope of a vertical line is undefined, not zero. Zero slope means the line is horizontal (flat). Undefined slope means the line is vertical (infinitely steep). These are opposite extremes, not the same thing.

• "Any two points give me a unique line." Almost — any two distinct points give a unique line. If the two points are the same point, infinitely many lines pass through that single point, and the slope formula gives 0/0, which is undefined.

• "The intercept form always works." It fails for lines through the origin (both intercepts are 0) and for lines parallel to an axis (one intercept is at infinity). The slope-intercept form is more general, and the general equation ax + by + c = 0 is the most general of all — it covers every line in the plane, including vertical lines.

• "Slope is always a number between -1 and 1." Slope can be any real number: 0, 1/2, -3, 100, -0.001. There is no bound. Steeper lines have slopes with larger absolute values. A slope of 100 means the line rises 100 units for every unit it moves to the right — nearly vertical.

• "Two lines with slopes m_1 and m_2 are perpendicular if m_1 = -m_2." This is only true when |m_1| = |m_2| = 1. The correct condition is m_1 \cdot m_2 = -1. For instance, lines with slopes 2 and -1/2 are perpendicular: 2 \times (-1/2) = -1. Lines with slopes 2 and -2 are not perpendicular: 2 \times (-2) = -4 \neq -1.

Going deeper

The general equation of a line

Every straight line in the plane can be written as:

where a, b, c are constants and a and b are not both zero. This is the most general form — it can represent every line, including vertical lines (set b = 0) and horizontal lines (set a = 0).

Every form you have seen is a special case:

• Slope-intercept: y = mx + c is the same as mx - y + c = 0, so a = m, b = -1.
• Intercept: x/a_0 + y/b_0 = 1 is the same as b_0 x + a_0 y - a_0 b_0 = 0.
• Normal: x\cos\alpha + y\sin\alpha = p is the same as \cos\alpha \cdot x + \sin\alpha \cdot y - p = 0.

The general equation ax + by + c = 0 is covered in detail in the article on General Equation of a Straight Line.

Distance from a point to a line

One of the most useful results in coordinate geometry: the perpendicular distance from a point (x_0, y_0) to the line ax + by + c = 0 is:

Derivation. Convert the line to normal form. Divide ax + by + c = 0 by \sqrt{a^2 + b^2}:

This is in the form x\cos\alpha + y\sin\alpha = p, where \cos\alpha = \frac{a}{\sqrt{a^2+b^2}}, \sin\alpha = \frac{b}{\sqrt{a^2+b^2}}, and p = \frac{-c}{\sqrt{a^2+b^2}}.

For any point (x_0, y_0), the signed distance from the origin to the line through (x_0, y_0) parallel to the given line is x_0\cos\alpha + y_0\sin\alpha. The signed distance from the origin to the given line itself is p. The perpendicular distance from (x_0, y_0) to the given line is the absolute difference:

This formula is used throughout coordinate geometry — in finding the distance between parallel lines, the area of a triangle, the equations of angle bisectors, and much more. It appears in virtually every JEE coordinate geometry problem.

The normal form and the dot product

The normal form x\cos\alpha + y\sin\alpha = p has a beautiful interpretation in terms of vectors.

The unit vector in the direction of the perpendicular from the origin is \hat{n} = (\cos\alpha, \sin\alpha). The position vector of any point P on the line is \vec{r} = (x, y). The equation x\cos\alpha + y\sin\alpha = p says:

In words: the projection of \vec{r} onto the direction of the perpendicular equals the perpendicular distance p. This is the dot-product interpretation of a line: a line is the set of all points whose position vectors have the same projection onto a fixed direction.

This interpretation generalises immediately to three dimensions. A plane in 3D is described by \vec{r} \cdot \hat{n} = p, where \hat{n} is the unit normal to the plane. The normal form of a line in 2D is the two-dimensional version of the equation of a plane.

Families of lines

Given a line L_1: a_1x + b_1y + c_1 = 0 and a line L_2: a_2x + b_2y + c_2 = 0, the equation

represents a family of lines passing through the intersection of L_1 and L_2, for different values of the parameter \lambda. Every line through the intersection (except L_2 itself) can be obtained by choosing the right \lambda.

This is a powerful technique for finding lines through an intersection point without actually computing the intersection. If you need a line through the intersection of 2x + 3y = 6 and x - y = 1 that also passes through the origin, write 2x + 3y - 6 + \lambda(x - y - 1) = 0 and set x = 0, y = 0: -6 - \lambda = 0, so \lambda = -6. The line is 2x + 3y - 6 - 6(x - y - 1) = 0, which simplifies to -4x + 9y = 0, or 4x = 9y.

Where this leads next

The equation of a straight line is the foundation of coordinate geometry. Once you have it, the next set of articles builds the full theory of straight lines and then extends to curves.

• General Equation of a Straight Line — the equation ax + by + c = 0, conditions for parallelism and perpendicularity, and the distance formula.
• Angle Between Two Lines and Related Conditions — how to find the acute angle between two intersecting lines, and the equations of angle bisectors.
• Coordinate Geometry Basics — the distance formula, section formula, and area formula that underpin everything in this article.
• Locus — the straight line as a locus, and the general idea of translating geometric conditions into equations.
• Graph Transformations and Translations — how shifting and stretching a graph changes the equation of a line or curve.