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What you will learn in Coordinate Geometry
1. The Cartesian Plane
Once a point is a pair of numbers, every geometric question becomes an arithmetic one.
2. Distance and Midpoint
The distance formula is Pythagoras with the triangle left implicit. This chapter draws it back in.
3. Dividing a Segment
Internal and external division are one formula, and the difference is a minus sign.
4. Gradient
Rise over run — and the one case where there is no run, which is not a very steep gradient.
5. Five Forms of a Line
Each form makes one thing obvious and hides the rest. Choosing well is the chapter's real skill.
6. Parallel and Perpendicular
The product of gradients is minus one exactly — and there is a case the product cannot describe at all.
7. Intersections, Angles and Distances
Three questions answered from the same coefficients, including whether three lines share a point.
8. The Circle
Two equations for one circle, and completing the square to get between them.
9. The Parabola
Equal distances to a point and a line. Trace the condition first; the equation comes after.
10. The Ellipse
A constant sum of distances to two foci — which is a loop of string and a pencil.
11. The Hyperbola
A constant difference instead of a sum, and the curve breaks into two branches.
12. One Family, One Number
Four curves, one equation, one slider — with both boundary values landed on exactly.
13. Locus Problems
A condition in words, a prediction, a trace, and only then an equation.
14. Choosing Better Coordinates
Move the axes, not the curve. A cross term is a sign you picked awkward axes.