Write true or false of each of the following statements. Also give reason for your answer.

(i) A tangent of a circle is that line which intersects the circle at two points.

(ii) A tangent XY touches a circle at a point P and Q is any other point on the tangent. If O is the centre of the circle then, OP = OQ.

(iii) Two tangent LM and XY are drawn respectively at two points P and Q on a circle. If PQ is the diameter than LM || XY.

(iv) The centre ‘O’ of a circle lies on another circle whose centre is ‘A’. If the circle with centre O passed through the points A and B such that AOB is a straight line, then the tangents drawn from the point B will pass through the points of intersection of the two points.

(i) The given statement is false. A tangent is a line which intersect the circle at exactly one point and the point of contact of tangent and circle is called as point of tangency.

The correct statement is as follows:

A secant of a circle is that line which intersects the circle at two points.

(ii) The given statement is false. Since O is the centre of the circle and hence OP is perpendicular to the tangent and the perpendicular is the smallest of all distances.

Therefore OP ≠ OQ.

A tangent XY touches a circle at a point P and Q is any other point on the tangent. If O is the centre of the circle then, OP < OQ.

(iii) The given statement is true. Since the tangent is perpendicular to the diameter, the two tangents LM and XY has to be parallel.

(iv) The given statement is true. Since AOB is a diameter, a semicircle is formed and the angle in a semicircle is always 90° or right angle.