Chapter 10 · Question 4

How many tangents can be drawn from a point to a circle when the point is (i) inside the circle, (ii) on the circle, and (iii) outside the circle? Explain with reasoning.

Answer Revealed

Direct Answer:

(i) Point inside the circle: No tangent can be drawn. Every line through an interior point is a secant (intersects the circle at two points). (ii) Point on the circle: Exactly one tangent can be drawn. By Theorem 10.1, only one line through a point on the circle can be perpendicular to the radius. (iii) Point outside the circle: Exactly two tangents can be drawn. These are the two lines from the external point that touch the circle.

Simple Explanation

If the point is inside the circle, you cannot draw a tangent — every line through it cuts the circle twice. If the point is on the circle, you can draw exactly one tangent. If the point is outside, you can draw exactly two tangents.

Exam-Ready Structure

(i) Point inside the circle: Zero tangents. Any line drawn through a point

P

inside the circle will intersect the circle at two distinct points, making it a secant. There is no line through an interior point that touches the circle at exactly one point. (ii) Point on the circle: Exactly one tangent. By Theorem 10.1, the tangent at that point is unique because it is the line perpendicular to the radius through the point of contact. Any other line through that point would intersect the circle at another point, making it a secant. (iii) Point outside the circle: Exactly two tangents. From an external point

P

, exactly two distinct lines can be drawn that just touch the circle. These two tangents are equal in length (by Theorem 10.2). This can be verified geometrically by drawing a circle through

P

with

OP

as diameter — it intersects the original circle at the two points of contact.

Key Points

Point inside circle: 0 tangents (all lines are secants)
Point on circle: 1 tangent (unique perpendicular to radius)
Point outside circle: 2 tangents (equal in length by Theorem 10.2)
At most 2 parallel tangents can be drawn to a circle

How many tangents can be drawn from a point to a circle when the point is (i) inside the circle, (ii) on the circle, and (iii) outside the circle? Explain with reasoning.