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Circles

In Figure, from an external point P, a tangent PT and a line segment PAB is drawn to a circle with centre O. ON is perpendicular on the chord AB. Prove that :

(i) PA . PB  =  PN²  – AN²

(ii) PN²  – AN²  =  OP²  –  OT²

(iii) PA.PB  =  PT²

Circles

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

Circles

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

Circles

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.

Circles

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

Circles

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

Circles

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

Circles

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.

Circles

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

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.