Circle Geometry and Algebraic Techniques
Circle intersections with straight lines follow predictable patterns - a line can intersect a circle twice, once (tangent), or not at all. Understanding this helps you predict how many solutions your equations will have.
To find a circle's equation, remember the three-step process: find the centre coordinates (a,b), calculate the radius using any given point, then substitute into x−a² + y−b² = r². For right-angled triangles inscribed in circles, the hypotenuse is always the diameter.
Algebraic fractions and polynomial division might look intimidating, but they follow the same rules as arithmetic division. The factor theorem is particularly useful - if x−a is a factor of f(x), then f(a) = 0.
Binomial expansion using Pascal's triangle gives you the coefficients for expressions like a+bⁿ. The index n+1 tells you which row of Pascal's triangle to use. For binomial estimation, you can approximate values like 0.975⁵ by treating it as 1−x⁵ where x = 0.025.
Remember: Tangents to circles are always perpendicular to the radius at the point of contact.