Coordinate Geometry & Proofs
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What you'll learn
The equation of a circle (centre/radius, completing the square, tangency via the discriminant), intersection of a line and a circle, and light coordinate-geometry proofs using gradients.
- A circle's equation comes straight from the distance-to-centre idea
- The radius is the distance from the centre to any point on the circle
- Completing the square turns the expanded (general) form into the standard form
- Finding where a line meets a circle: substitute, then solve the quadratic
- A line is tangent to a circle exactly when that quadratic has ONE repeated root
- Gradient and distance are the two basic tools behind every coordinate proof
- Equal gradients prove collinearity; gradients multiplying to \(-1\) prove a right angle