Coordinate geometry
Straight lines
- Gradient of the line through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.
- Line equations: $y = mx + c$, or $y - y_1 = m(x - x_1)$.
- Parallel lines have equal gradients; perpendicular lines have gradients that multiply to $-1$.
Practice
What is the gradient of the line through (2, 3) and (6, 11)?
m = (11 − 3) / (6 − 2) = 8/4 = 2.
Practice
A line has gradient 2. What is the gradient of a line perpendicular to it?
Perpendicular gradients multiply to −1, so the gradient is −1/2 = −0.5.
Circles
- A circle, centre $(a, b)$, radius $r$: $(x - a)^2 + (y - b)^2 = r^2$.
- An expanded form (e.g. $x^2 + y^2 - 6x + 10y - 27 = 0$) → complete the square in $x$ and $y$ to find the centre and radius.
- A tangent touches the circle once and is perpendicular to the radius there.
Practice
The circle (x − 3)² + (y + 2)² = 25 has what radius?
r² = 25, so r = 5; the centre is (3, −2).
Practice
A tangent to a circle is perpendicular to the radius at the point where it touches.
This right-angle fact solves most circle problems.
You've got it
Key idea
- gradient $m = \dfrac{y_2 - y_1}{x_2 - x_1}$; perpendicular gradients multiply to $-1$
- circle: $(x-a)^2 + (y-b)^2 = r^2$, centre $(a,b)$, radius $r$
- a tangent is perpendicular to the radius at the point of contact