Circles

Board: CBSE | Class: Class 10

Comprehensive study notes for Circles by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Tangent to a Circle

A tangent is a line that touches the circle at exactly one point (the point of contact). A line intersecting the circle at two points is a secant.

Radius-Tangent Theorem

The tangent at any point of a circle is perpendicular to the radius through the point of contact. If OP is radius and PT is tangent at P, then OP ⊥ PT.

Number of Tangents

From a point inside a circle: no tangent. From a point on the circle: exactly one tangent. From a point outside the circle: exactly two tangents.

Tangent Lengths

The lengths of tangents drawn from an external point to a circle are equal. If PA and PB are tangents from P to circle with center O, then PA = PB.

Tangent-Secant Theorem

If a tangent and a secant intersect at an external point: (tangent length)² = (secant segment length)(external part).

Important Formulas

Tangent-Radius	OP ⊥ PT (radius ⊥ tangent)
Equal Tangents	PA = PB (tangents from same external point)
Tangent-Secant	PT² = PA × PB
Number of Tangents	Inside: 0, On: 1, Outside: 2

Solved Examples

Example 1: A tangent PT touches circle at P. OP = 5 cm, PT = 12 cm. Find distance from O to T.

Solution: Since OP ⊥ PT, OT² = OP² + PT² = 25 + 144 = 169. OT = 13 cm.

Example 2: Two tangents PA and PB are drawn from P to a circle. If ∠APB = 60°, find ∠AOB.

Solution: OA ⊥ PA, OB ⊥ PB. In quadrilateral AOBP: ∠OAP = ∠OBP = 90°. ∠AOB = 360 – (90+60+90) = 120°.

Example 3: Find the length of tangent from point 13 cm from center of circle of radius 5 cm.

Solution: PT² = OP² – r² = 169 – 25 = 144. PT = 12 cm.

Practice Questions

1. Find the radius of a circle if a tangent of length 8 cm is drawn from a point 10 cm from center.
2. Prove that tangents drawn from an external point are equal in length.
3. A tangent PQ touches circle at Q. OP = 25 cm, PQ = 24 cm. Find radius.
4. Two concentric circles have radii 5 cm and 3 cm. Find chord length of larger circle tangent to smaller.
5. In a circle of radius 7 cm, tangent PT = 24 cm. Find OP.

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