Question

P is point lying outside a cicle. 3 lines are drawn from P to the cicle so that they intersect the circle at 6 points as shown in the figure.

$P_1,P_2,P_3$ are points formed by intersections of tangents at A and B, C and D and E and F respectively.

• A. $P_1,P_2,P_3$ are always collinear
• B. $P_1,P_2,P_3$ may be collinear
• C. $P_1,P2,P_3$ cannot be collinear
• D. No relation possible for $P_1,P_2,P_3$

Answer 1

The correct option is A

$P_1,P_2,P_3$ are always collinear

You can try taking any number of lines from P which intersects the circle. All the tangents from such points intersect on a line. This line by definition is called the polar of the point and the point is called the pole of the polar.

This is same as the case two that we discussed when the point P is outside the circle. When point is outside as in the figure, any number of lines can be drawn to

the circle which can intersect the circle at 2 points, say x and y. The tangents drawn from x and y meets at T. The locus of all such $T_s$ is called the polar of point P with respect to the circle.

Therefore by definition of polar of a point, all such points of intersection forms a straight line.