NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 9: Circles

Exercise 9.2

Out of 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 \(P\) to a circle with centre \(O\). Prove that \(QORP\) is a cyclic quadrilateral.

From an external point \(B\) of a circle with centre \(O\), two tangents \(BC\) and \(BD\) are drawn such that \(\angle DBC=120^\circ\). Prove that \(BC+BD=BO\) (equivalently, \(BO=2\,BC\)).

Prove that the centre of a circle touching two intersecting straight 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 \(AB=CD\).

In Question 5 above, if the 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 \(AB=CD\).

A chord \(PQ\) of a circle is parallel to the tangent 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 every chord that is parallel to the tangent at \(A\).