Coordinate geometry of lines

Equation of a line from two points, general form ax + by + c = 0, parallel and perpendicular lines, distance, midpoint and collinear points.

General form

The general form of a line is $ax + by + c = 0$ with whole-number $a$, $b$, $c$ and $a > 0$. For example, $y = \tfrac{4}{3}x + 1$ becomes $4x - 3y + 3 = 0$.

Example 1●●●

Find the line perpendicular to $3x + 4y = 12$ through $(0, 1)$.

$3x + 4y = 12$ has $m = -\tfrac{3}{4}$, so the perpendicular gradient is $\tfrac{4}{3}$.

$y = \tfrac{4}{3}x + 1$, or $4x - 3y + 3 = 0$.

Example 2●●

Are $(1, 1)$, $(2, 3)$ and $(4, 7)$ collinear (on one line)?

Gradients: $\dfrac{3 - 1}{2 - 1} = 2$ and $\dfrac{7 - 3}{4 - 2} = 2$.

Equal gradients through a shared point, so yes.

Flashcards

General form

$ax + by + c = 0$

Collinear test

Equal gradients between pairs of points

Gradient of $ax + by + c = 0$

$-\dfrac{a}{b}$

Practice

Line through $(-1, 2)$ and $(3, 10)$: $y = $ ?
Gradient perpendicular to $3x + 4y = 12$?
Midpoint of $(-2, 7)$ and $(6, -1)$?
Exact distance from $(-2, 7)$ to $(6, -1)$? Type like 8sqrt(2).
Are $(1, 1)$, $(2, 3)$, $(4, 7)$ collinear?
Gradient of $2x + 5y - 10 = 0$?
$y = \tfrac{4}{3}x + 1$ in general form is:
$y = kx + 2$ is perpendicular to $y = x - 5$. Find $k$.