TMUA coordinate geometry: translate geometry into algebra cleanly.

The high-value circle moves

Complete the square to identify a circle's centre and radius; use a perpendicular radius for tangent geometry; subtract two circle equations to obtain the common chord line; and use the distance from a centre to a line when the geometry is easier than substitution.

When a line is substituted into a circle, the resulting quadratic has a geometric meaning: two real roots means a secant, one repeated root means tangency, and no real roots means no intersection.

Parameter questions

A parameter may appear in the gradient, intercept, centre or radius. Instead of solving coordinates explicitly, ask what geometric event marks the boundary of the allowed values. In many problems that boundary is tangency, so the discriminant becomes zero or a distance becomes exactly equal to the radius.