Question

In the given figure, a line intersects a circle in two points A and B. Now a point P, as shown in the figure, is fixed and the line is rotated about P in both directions until the points A and B coincide. How many times will they coincide?

• A. 1
• B. 2
• C. Infinite

Answer 1

The correct option is B 2


When the line is rotated clockwise about P, as shown in the figure below, the points A and B get closer to each other until they coincide at point T as shown

When the line is rotated anticlockwise, the points A and B initially go further away from each other until they become the ends of a diameter (denoted by A’ and B’), in which case the line would pass through the centre of the circle (denoted by O). Then, the points A and B would come closer to each other until they coincide at point S.

Hence, there are two places where the points A and B would coincide, and those are points S and T as shown in the figures above.
The tangent is a special case of secant, in which the points of intersection of secant and the circle coincide. In this case, the secant, when rotated about the external point attains the special case twice, which are the tangents possible from that external point.