Q.Find the centre and radius of the circle $x^2 + y^2 - 8x + 10y - 12 = 0$.

Properties of Circles – From Intuition to Precision

A circle is the set of all points that lie at a fixed distance from a single point. That single point is the centre, and the fixed distance is the radius. If you tie a string to a pencil, hold the other end fixed, and swing the pencil around, the curve it traces is a circle. Every point on that curve is exactly one string-length away from your finger.

That simple idea — constant distance from a centre — is the seed from which every property of a circle grows.

The Basic Parts

Before we talk about properties, name the pieces.

• Centre ($O$): the fixed point.
• Radius ($r$): the distance from centre to any point on the circle.
• Diameter ($d$): a line segment through the centre, joining two points on the circle. $d = 2r$.
• Chord: any line segment joining two points on the circle. The diameter is the longest chord.
• Arc: a part of the circumference (the curved boundary).
• Sector: the region between two radii and the arc they cut off.
• Segment: the region between a chord and the arc it cuts off.

Property 1: Equal Chords Are Equidistant from the Centre

Take a circle. Draw two chords of equal length. Drop perpendiculars from the centre to each chord. Those perpendicular distances will be equal. Conversely, if two chords are at the same distance from the centre, they are equal in length.

Why? The perpendicular from the centre to a chord bisects the chord. So each chord is split into two equal halves. If the chords are equal, the right triangles formed by the radius, the half-chord, and the perpendicular are congruent — hence the perpendicular distances match.

Property 2: The Angle Subtended by a Chord at the Centre Is Twice the Angle Subtended at Any Point on the Circumference

This is the most powerful property of circles. Draw a chord $AB$. Let $O$ be the centre. Pick any point $P$ on the circumference (on the same side of $AB$). Then:

The central angle is always double the inscribed angle.

Why? Join $OA$, $OB$, $OP$. You get two isosceles triangles. The exterior angle of a triangle equals the sum of the two opposite interior angles. Work through the geometry — the doubling falls out naturally.

Property 3: Angles in the Same Segment Are Equal

If you take a chord $AB$ and pick two different points $P$ and $Q$ on the same arc (the same side of $AB$), then:

Both are half of the same central angle $\angle AOB$, so they must be equal. This is a direct consequence of Property 2.

Property 4: The Tangent Is Perpendicular to the Radius at the Point of Contact

A tangent is a line that touches the circle at exactly one point. At that point, the radius drawn to the point is perpendicular to the tangent. …