CBSE Class 10 Mathematics Circles Notes Set 07

Terms Related to a Circle

A secant is a line that cuts the circle at two points. A tangent is a line that meets the circle at exactly one point, called the point of contact. A chord through the centre is a diameter. A circle has no tangent through a point inside it, exactly one tangent at a point on it, and exactly two tangents from a point outside it.

Theorem 1: Tangent and Radius

The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.
Why: every other point of the tangent lies outside the circle, so the radius to the point of contact is the shortest distance from the centre to the line, and the shortest distance is the perpendicular.
Converse: a line drawn through the end of a radius and perpendicular to it is a tangent.

Theorem 2: Equal Tangents

The two tangent segments drawn from an external point to a circle are equal in length.
Proof idea: in right triangles OAP and OBP (A and B are the points of contact), OA = OB, OP is common and ∠OAP = ∠OBP = 90°, so the triangles are congruent by RHS and PA = PB.
Consequences: OP bisects the angle APB and the angle AOB, and OP is the perpendicular bisector of the chord AB.

Key Results to Remember

1. Length of tangent from a point at distance d from the centre of a circle of radius r: \( \sqrt{d^2 - r^2} \).
2. Angle between the two tangents and the angle between the two radii at the points of contact are supplementary: ∠APB + ∠AOB = 180°.
3. Tangents drawn at the ends of a diameter are parallel.
4. A parallelogram circumscribing a circle is a rhombus, and a rectangle circumscribing a circle is a square.
5. In a quadrilateral ABCD circumscribing a circle, AB + CD = BC + DA.
6. For a right triangle with legs a, b and hypotenuse c, the incircle radius is \( r = \frac{a + b - c}{2} \).
7. Two circles can have 4, 3, 2, 1 or 0 common tangents when they are separate, touch externally, intersect, touch internally, or one lies inside the other, respectively.
8. Perpendicular from the centre to a chord bisects the chord; equal chords are equidistant from the centre.

Common Tangents of Two Circles

For circles with radii r₁ and r₂ whose centres are d apart:
Length of direct (external) common tangent = \( \sqrt{d^2 - (r_1 - r_2)^2} \).
Length of transverse (internal) common tangent = \( \sqrt{d^2 - (r_1 + r_2)^2} \).

Worked Examples

Example 1. Two tangents PA and PB are drawn from P to a circle with centre O, and ∠APB = 70°. Find ∠OAB.
Solution: ∠AOB = 180° − 70° = 110°. Triangle OAB is isosceles (OA = OB), so ∠OAB = (180° − 110°) ÷ 2 = 35°.

Example 2. A quadrilateral ABCD circumscribes a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm, find AD.
Solution: AB + CD = BC + AD, so 6 + 4 = 7 + AD and AD = 3 cm.

Example 3. The tangent length from a point P to a circle is 24 cm and OP = 25 cm. Find the radius.
Solution: r² = OP² − PT² = 625 − 576 = 49, so r = 7 cm.

Example 4. In a triangle ABC the incircle touches BC, CA and AB at D, E and F. If AB = 12 cm, BC = 8 cm and CA = 10 cm, find AF, BD and CE.
Solution: Let AF = AE = x, BF = BD = y, CD = CE = z. Then x + y = 12, y + z = 8, x + z = 10. Adding gives x + y + z = 15, so z = 3, x = 7, y = 5. Thus AF = 7 cm, BD = 5 cm, CE = 3 cm.

Example 5. Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
Solution: Let tangents at P and Q meet at T. TP = TQ, so triangle TPQ is isosceles and ∠TPQ = ∠TQP.