Ch 11Circle — Class 12 Mathematics, concept-first.

Geometry is one of the oldest branches of mathematics, with roots that stretch back to ancient Egypt before flourishing further in Greece, India and China; its systematic, proof-based development is u…

A circle is simply the set of every point in a plane that sits at one fixed distance, the radius , from one fixed point, the centre .

Often a problem does not hand you the centre and radius directly — instead it describes the circle through the points it passes through. Two classic situations come up again and again.

The Cartesian equation describes a circle implicitly — it tells you which pairs lie on the curve, but not how to sweep through them one at a time.

Given a circle and any point in the plane, a very useful quantity is obtained by simply substituting the point's coordinates into :

A straight line can relate to a circle in exactly three ways: it can cut through the circle at two distinct points (a secant), just graze it at exactly one point (a tangent), or miss it entirely.

At any point on a circle, the normal is the line through that is perpendicular to the tangent at . Because a tangent is always perpendicular to the radius drawn to the point of contact, the normal at…

When a point lies outside a circle, two distinct tangent lines can be drawn from to the circle, touching it at two points, say and .

The chord-of-contact idea generalises into one of the more elegant constructions in the chapter. Let be a circle and any point in the plane other than the centre.

So far every idea has concerned a single circle. It's also useful to know how two circles can sit relative to one another, since this governs how many common tangent lines they share — a line that is…

Section 1.7 found the chord of contact joining the two points where tangents from an external point touch a circle.

Sample papers and previous-year board questions for this subject.