Which of the following statements is true?

(a) The tangents drawn at the end points of a chord of a circle are parallel.
(b) From a point P on the exterior of a circle, only two secants can be drawn through P to the circle.
(c) From a point P in the plane of a circle, two tangents can be drawn to the circle.
(d) The perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

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Solution

(d) The perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Let O be the centre of the given circle.

A tangent PR has been drawn touching the circle at point P.
Draw QP ⊥ RP at point P, such that point Q lies on the circle. $∠$ OPR = 90° (Since radius ⊥ tangent)
Also, $∠$ QPR = 90° (Given)
∴ $∠$ OPR = $∠$ QPR
Now, the above case is possible only when the centre O lies on the line QP.
Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

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Similar questions

Q. Which of the following statements is not true?
(a) If a point P lies inside a circle, no tangent can be drawn to the circle passing through P.
(b) If a point P lies on a circle, then one and only one tangent can be drawn to the circle at P.
(c) If a point P lies outside a circle, then only two tangents can be drawn to the circle from P.
(d) A circle can have more than two parallel tangents parallel to a given line.

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(b) If a point P lies on the circle, then one and only tangent can eb drawn to the circle at P.

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