Exemplar Exercise 9.3 (Circles)

Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.

Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.

If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that ∠DBC = 120°, prove that BC + BD = BO, i.e., BO = 2BC.

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

In Fig. 9.13, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.

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In Question 5 above, if radii of the two circles are equal, prove that AB = CD.

In Fig. 9.14, common tangents AB and CD to two circles intersect at E. Prove that AB = CD.

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A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A.