Chapter 10

Circles

Definition: A circle is a collection of points in a plane which are at a constant distance from a fixed point.

Here, O is the fixed point, so O is the centre of the circle. The distance of the points A, B, C, D, E, F etc from O are same and this fixed distance is the radius of the circle.

Secant: A line which intersects a circle in two distinct points is called a secant of the circle.

Chord: A line segment joining two points on the circle is called chord of the circle. Or A part of secant interior to the circle is called the chord of the circle.

Tangent: If a line intersects a circle in only one point, then the line is called tangent of the circle and the intersecting point is called the point of contact of the tangent.

In this figure is the secant of the circle. Line segments AB and PQ are the chords of circle. Line m is tangent to the circle and point X is the point of contact.

Remark: The point of contact is only one point which is common to the circle and tangent, other points on the tangent lie outside to the circle. So, the point of contact is nearest to the center of the circle.

Some Properties of the Tangent to the Circle:

(1) A tangent to the circle is perpendicular to the radius through the point of contact.

(2) From a point on the circle only one tangent can be drawn.

(3) From a point outside to the circle, two tangents can be drawn.

(4) No tangent can be drawn from the point interior to the circle.

(5) The lengths of tangents drawn from an external point to a circle are equal.

PX = PY

(6) In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

Here, AB is the chord of outer circle touches the smaller at P. P is the point of contact of the tangent to the smaller circle.

Thus, AP = PB