The angle between two tangents drawn from an external point to a circle is 60°. What is the angle subtended by the chord connecting their points of contact at the center?
120°
Let \(P\) be the external point and let the two tangents touch the circle at \(A\) and \(B\). Then \(\angle APB = 60^\circ\).
The radii \(OA\) and \(OB\) are perpendicular to the tangents at the points of contact:
\(\angle OAP = \angle OBP = 90^\circ\)
In the quadrilateral \(OAPB\), the four angles must sum to 360°:
\(\angle AOB + \angle APB + \angle OAP + \angle OBP = 360^\circ\)
\(\angle AOB + 60^\circ + 90^\circ + 90^\circ = 360^\circ\)
\(\angle AOB = 360^\circ - 240^\circ = 120^\circ\)
The chord \(AB\) subtends \(\angle AOB = 120^\circ\) at the centre — option (2).
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Important Questions from Circles, Chords and Tangents
If a tangent to a circle from a point P meets the circle at A with AP = 15 cm. Given that the radius of the circle is 8 cm, find the distance of P from the centre of the circle.
In a circle with a radius of 10 cm. XY and PQ are two parallel chords 12 cm and 16 cm in length, respectively. The two chords are situated on the opposite sides of the centre. The distance between the chords is:
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Let \( C_1 \) and \( C_2 \) be two circles which do not externally touch and intersect each other and \( O_1 \), and \( O_2 \) be the centers of the circles, respectively. Let AB be the common transverse tangent to the circles such that P, Q are the points of tangency respectively to \( C_1 \), \( C_2 \). Let R be the point of intersection of \( O_1 O_2 \) and AB. If \( \angle PO_1R = 60^\circ \), find \( \angle QO_2R \) and \( \angle QRO_2 \) respectively.
