9709 Pure 1 · 1.1 Quadratics · 1.3 Coordinate Geometry
Solving a line and a curve together is one question in disguise: where do they touch? Slide the line and watch the algebra keep up. The discriminant of the quadratic you land on tells you how many answers you are going to get, before you work any of them out. Tangent at a point is for 1.3: drag a point around a circle and see why the tangent there is perpendicular to the radius.
drag on the graph to move the line
Try it yourself. Move the slider until you think the line is exactly a tangent, then check. Tangency is the Δ = 0 case, and it is worth being able to feel where it happens.
The exam method. Find the gradient of CP from the center and the point. Flip it and change the sign to get the tangent's gradient. Then use y − y₁ = m(x − x₁) through P and tidy into ax + by + c = 0.
Two quick checks. P has to satisfy the circle's equation. Substituting your tangent into the circle should leave a quadratic with Δ = 0.
Why there are usually two answers. A straight line usually crosses a curve in two places, so a pair of simultaneous equations usually has two answer pairs. Stopping after one x-value is half an answer, and every x owes you the matching y that sits directly above or below it on the line.
Where the discriminant comes in. Substituting the line into the curve always leaves an ordinary quadratic. Δ > 0 gives two crossings, Δ = 0 gives a tangent, and Δ < 0 means the line misses, which is the algebra's way of saying there is no real answer to find.