Pure · Coordinate geometry
Chapter 1 · 4
The idea
Lines and circles — tangents, chords, and intersections
Where a line meets a circle (substitute, then read the discriminant), and the three circle facts that solve coordinate problems: tangent ⊥ radius, the perpendicular bisector of a chord passes through the centre, and the angle in a semicircle is 90°.
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In this lesson — start anywhere
Pure · Coordinate geometry
Lines and circles — tangents, chords, and intersections
Where a line meets a circle (substitute, then read the discriminant), and the three circle facts that solve coordinate problems: tangent ⊥ radius, the perpendicular bisector of a chord passes through the centre, and the angle in a semicircle is 90°.
Why it works
Substitute, then read the discriminant
Where does a line meet a circle? Substitute the line into the circle's equation. You get a quadratic in , and its discriminant tells you the geometry before you even solve it:Proving a tangent
This is the way to prove a line is a tangent, or to find the value of a constant that makes it one: substitute, collect the quadratic, set , solve. Having found , always substitute back into the line (the simpler equation) to get — a coordinate needs both numbers, and the circle's equation would offer you a spurious second .Keep reading — free
The rest of the explanation, plus 3 worked examples you step through move by move.
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