Coordinate Geometry: Lines, Circles & Polar Coordinates

The Distance Between Two Points

$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Worked example: distance between A(2,3) and B(5,7) $= \sqrt{(5-2)^2+(7-3)^2} = \sqrt{9+16} = \sqrt{25} = 5$.

You can also run this formula backwards: if you're told the distance and one coordinate is unknown, set up the equation and solve. E.g. if the distance between (2,5) and (x,9) is 5: $(x-2)^2+16 = 25 \Rightarrow (x-2)^2 = 9 \Rightarrow x-2 = \pm 3 \Rightarrow x = 5 \text{ or } -1$.

The Midpoint Formula

$M = \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$

Worked example: midpoint of A(2,4) and B(6,8) $= \left(\dfrac{2+6}{2}, \dfrac{4+8}{2}\right) = (4,6)$.

This also runs backwards: if you know the midpoint and one endpoint, you can solve for the missing coordinate of the other endpoint. E.g. if the midpoint of P(−3,y) and Q(5,7) is M(1,4): $\dfrac{y+7}{2} = 4 \Rightarrow y = 1$.

Slope

$\text{Slope} = \dfrac{y_2-y_1}{x_2-x_1}$

Worked example: the slope through (1,2) and (2,4) $= \dfrac{4-2}{2-1} = 2$.

Parallel and Perpendicular Lines

• Parallel lines have the exact same slope. A line parallel to y=2x+3 also has slope 2.
• Perpendicular lines have slopes that are negative reciprocals of each other (multiply to −1). A line perpendicular to y=(1/3)x+1 has slope −3.

Collinearity — Are Three Points on the Same Line?

Three points are collinear if the slope between any two pairs of them is identical. Worked example: for A(2,3), B(4,7), C(k,11) to be collinear, the slope of AB must equal the slope of BC: $\dfrac{7-3}{4-2} = 2$, so $\dfrac{11-7}{k-4} = 2 \Rightarrow k-4 = 2 \Rightarrow k = 6$.

The Equation of a Circle

$(x-h)^2 + (y-k)^2 = r^2$

where (h,k) is the centre and r is the radius.

Worked example: a circle with radius 5 and centre (−3,2): $(x-(-3))^2 + (y-2)^2 = 5^2 \Rightarrow (x+3)^2 + (y-2)^2 = 25$.

Converting Between Polar and Cartesian Coordinates

• Polar → Cartesian: $x = r\cos\theta$, $y = r\sin\theta$.
• Cartesian → Polar: $r = \sqrt{x^2+y^2}$, and θ is found from the point's quadrant.

Worked example: the point (5, 60°) in polar form → x = 5cos60° = 2.5, y = 5sin60° ≈ 4.33, so Cartesian ≈ (2.5, 4.3). Going the other way, the point (−3,0) has r = 3, and since it sits on the negative x-axis, θ = 180°.

Why This Matters for Your Exams

Every formula on this page takes the same two or three coordinate pairs as input — the skill being tested is picking the right formula for what's asked (distance vs midpoint vs slope), not the arithmetic itself, which is always straightforward once you've chosen correctly.