Coordinate Geometry: Circles & Curves

At a glance 11 formulas

1. 1Centre and radius$(x - h)^2 + (y - k)^2 = r^2$$\text{centre } (h, k),\ \text{radius } r$
2. 2General form$x^2 + y^2 + 2gx + 2fy + c = 0$$\text{centre } (-g, -f),\ r = \sqrt{g^2 + f^2 - c}$
3. 3Circle on a diameter$(x - a)(x - c) + (y - b)(y - d) = 0$$\text{ends } (a, b), (c, d);\ 90^\circ \text{ at } P \Rightarrow P \text{ on it}$
4. 4Where (x₁, y₁) lies$S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$$S_1 < 0 \text{ inside},\ = 0 \text{ on},\ > 0 \text{ outside}$
5. 5Line d from centre$d < r \text{ cuts},\ d = r \text{ touches}$$d > r \text{ misses};\ \text{chord} = 2\sqrt{r^2 - d^2}$
6. 6Nearest and farthest point$\text{nearest} = |d - r|,\ \text{farthest} = d + r$$d = \text{distance from the point to the centre}$
7. 7Tangents from P$\text{length} = \sqrt{d^2 - r^2} = \sqrt{S_1}$$d = P \text{ to centre};\ \sin\tfrac{\theta}{2} = \tfrac{r}{d},\ \theta \text{ at } P$
8. 8Two circlesCentres $d$ apart: $d > r_1 + r_2$ gives 4 common tangents, $d = r_1 + r_2$ gives 3, in between gives 2, $d = |r_1 - r_2|$ gives 1, and less gives none.
9. 9Circle and the axesWith centre $(h, k)$, the circle touches the x-axis when $r = |k|$ and the y-axis when $r = |h|$; it stays clear of both when $r$ is less than both.
10. 10Fixed ratio of distancesIf $PA : PB = k$ with $k \ne 1$, $P$ traces a circle; the two points dividing $AB$ in $k : 1$, inside and outside, are the ends of a diameter.
11. 11Circle meets a curveSubstitute to get one equation in $y$ (or $x$). Each root gives two points, or one if the other coordinate is 0 (a touch), or none if its square is negative.