Straight Lines: Slope and Forms of Equation | NIMCET

Quick answer: The slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta$, where θ is its inclination. The most used forms are slope-intercept $y = mx + c$, point-slope $y - y_1 = m(x - x_1)$, intercept $\frac{x}{a} + \frac{y}{b} = 1$, and general $ax + by + c = 0$ with slope $-\frac{a}{b}$. Lines are parallel if $m_1 = m_2$ and perpendicular if $m_1 m_2 = -1$.

Slope

Parallel lines: $m_1 = m_2$. Perpendicular lines: $m_1 m_2 = -1$ (or one horizontal and the other vertical).

Collinearity by slope: A, B, C are collinear if slope AB = slope BC.

Forms of the equation of a line

From general form to other forms

For $ax + by + c = 0$:

To normal form: divide by $\pm\sqrt{a^2 + b^2}$, choosing the sign that makes the right-hand side positive.

Lines parallel and perpendicular to a given line

Find k by substituting the given point.

Worked examples

Example 1: two-point form

Find the line through $(1, 2)$ and $(3, 8)$.

Step 1: $m = \frac{8 - 2}{3 - 1} = 3$.

Step 2: $y - 2 = 3(x - 1)$, so $y = 3x - 1$.

Example 2: intercept form

A line cuts intercepts 4 and −3 on the axes. Find its equation.

$\frac{x}{4} + \frac{y}{-3} = 1$, so $3x - 4y = 12$.

Example 3: perpendicular line through a point

Find the line through $(2, -1)$ perpendicular to $3x + 4y = 7$.

Step 1: Perpendicular form: $4x - 3y + k = 0$.

Step 2: Substitute $(2, -1)$: $8 + 3 + k = 0$, so $k = -11$.

Answer: $4x - 3y = 11$.

Example 4: normal form

Write $\sqrt{3}x + y = 4$ in normal form.

Divide by $\sqrt{3 + 1} = 2$: $\frac{\sqrt{3}}{2}x + \frac{1}{2}y = 2$, that is, $x\cos 30^\circ + y\sin 30^\circ = 2$. So $p = 2$ and $\alpha = 30^\circ$.

Summary table

Exam tips

• Use the "swap and change sign" trick for perpendicular lines: $ax + by \to bx - ay$.
• For a line with equal intercepts, use $x + y = a$; for intercepts equal in size but opposite in sign, $x - y = a$.
• The area of the triangle a line makes with the axes is $\frac{1}{2}\lvert ab\rvert$ for intercepts a and b.

Next, see angle between lines and distance of a point, or browse the NIMCET Coordinate Geometry and Vectors chapter.

Common mistakes

• Writing the slope of $ax + by + c = 0$ as $\frac{a}{b}$ instead of $-\frac{a}{b}$.
• Using $m_1 m_2 = -1$ when one line is vertical.
• Forgetting the sign of a negative intercept in the intercept form.
• Leaving p negative in the normal form.

Frequently Asked Questions

What is the slope of the line ax + by + c = 0?

The slope is −a/b.

When are two lines perpendicular?

When the product of their slopes is −1, or when one is horizontal and the other vertical.

What is the intercept form of a line?

x/a + y/b = 1, where a and b are the x- and y-intercepts.

What is the normal form of a line?

x cos α + y sin α = p, where p is the perpendicular distance from the origin and α is the angle that perpendicular makes with the x-axis.

How do you find a line perpendicular to ax + by + c = 0?

Write bx − ay + k = 0 and find k from the given point.