Further equations of lines

Point–gradient form, the line through two points, parallel and perpendicular lines through a point, and testing whether three lines are concurrent.

Point–gradient form

$$y - y_1 = m(x - x_1)$$

This is the quickest way to write a line when you know one point and the gradient.

Example 1●

Find the line through $(2, -3)$ with gradient $4$.

$y + 3 = 4(x - 2)$

$y = 4x - 11$

The line through two points

Example 2●●

Find the line through $A(-1, 5)$ and $B(3, -3)$.

$m = \dfrac{-3 - 5}{3 - (-1)} = -2$

$y - 5 = -2(x + 1)$, so $y = -2x + 3$.

Use either point.

Line through two points, and its perpendicular

Drag A and B. The dashed line is perpendicular to AB and passes through A.

Try this

Make $AB$ horizontal. What does the perpendicular become?

Example 3: perpendicular through a point●●

Find the line through $(4, 1)$ perpendicular to $2x - 3y + 6 = 0$. Give it in general form.

The given line has gradient $\tfrac23$, so the perpendicular gradient is $-\tfrac32$.

$y - 1 = -\tfrac32(x - 4)$

$2y - 2 = -3x + 12$, so $3x + 2y - 14 = 0$.

Multiply by 2 and rearrange.

Intersections and concurrent lines

Lines are concurrent if they all pass through one point. To test three lines, find where two of them meet (simultaneous equations), then check that point in the third.

Example 4●●●

Are $y = x + 1$, $y = -x + 5$ and $y = 3x - 3$ concurrent?

First two: $x + 1 = -x + 5$, so $x = 2$, $y = 3$.

Third line at $x = 2$: $y = 3(2) - 3 = 3$. ✓

All three pass through $(2, 3)$, so they are concurrent.

Flashcards

Point–gradient form

$y - y_1 = m(x - x_1)$

Line through two points

Find $m$, then use point–gradient form.

Concurrent

Three or more lines through one point.

Practice

The line through $(1, -2)$ with gradient $-5$ is $y = -5x + b$. Find $b$.
The line through $(0, 2)$ and $(4, 10)$ is:
A line perpendicular to $y = -\tfrac13 x + 2$ through the origin is:
Where do $y = 2x - 1$ and $y = -x + 5$ meet? Enter the $x$-coordinate.
The line through $(3, 4)$ parallel to $y = 5x$ is:
Lines are concurrent when they:
For which $k$ does $y = kx + 7$ pass through $(2, 3)$?
The vertical line through the intersection of $y = x$ and $y = 6 - x$ is: