TMUA Coordinate Geometry

Coordinate geometry on TMUA Paper 1 is mostly about translating a geometric condition (parallel, perpendicular, tangent, intersection, distance) into an algebraic one and then solving. The geometry itself is GCSE-level; the test is the translation.

• Gradients. Two lines are parallel iff their gradients are equal; perpendicular iff their gradients multiply to $-1$. The perpendicular rule is the one most often hidden inside a question that doesn’t mention perpendicularity.
• Distance and midpoint. $|PQ| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$; midpoint of $PQ$ is the coordinate-average. The midpoint of a chord of a circle lies on the radius through the centre.
• Circle equation. $(x - a)^2 + (y - b)^2 = r^2$ with centre $(a, b)$ and radius $r$. From the expanded form $x^2 + y^2 + Dx + Ey + F = 0$, centre is $(-D/2, -E/2)$ and radius is $\sqrt{D^2/4 + E^2/4 - F}$.
• Line meets circle. Substitute the line into the circle equation; the resulting quadratic in $x$ has discriminant zero for a tangent, positive for two intersection points, negative for none.

The move. A condition about how two shapes touch or cross is always a discriminant condition once you’ve substituted one into the other. Don’t try to compute intersections explicitly when the question only needs the sign of the discriminant.