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

Exercise 9.2

If a chord \(AB\) subtends an angle of \(60^\circ\) at the centre of a circle, then the angle between the tangents at \(A\) and \(B\) is also \(60^\circ\). State True/False and justify.

The length of a tangent from an external point on a circle is always greater than the radius of the circle. State True/False and justify.

The length of the tangent from an external point \(P\) on a circle with centre \(O\) is always less than \(OP\). State True/False and justify.

The angle between two tangents to a circle may be \(0^\circ\). State True/False and justify.

If the angle between two tangents drawn from a point \(P\) to a circle of radius \(a\) and centre \(O\) is \(90^\circ\), then \(OP = a\sqrt{2}\). State True/False and justify.

If the angle between two tangents drawn from a point \(P\) to a circle of radius \(a\) and centre \(O\) is \(60^\circ\), then \(OP = a\sqrt{3}\). State True/False and justify.

The tangent to the circumcircle of an isosceles triangle \(\triangle ABC\) at \(A\) (where \(AB=AC\)) is parallel to \(BC\). State True/False and justify.

If a number of circles touch a given line segment \(PQ\) at a point \(A\), then their centres lie on the perpendicular bisector of \(PQ\). State True/False and justify.

If a number of circles pass through the end points \(P\) and \(Q\) of a line segment \(PQ\), then their centres lie on the perpendicular bisector of \(PQ\). State True/False and justify.

\(AB\) is a diameter of a circle and \(AC\) is a chord such that \(\angle BAC = 30^\circ\). If the tangent at \(C\) meets \(AB\) produced at \(D\), then \(BC = BD\). State True/False and justify.