Number of Tangents from a Point

How Many Tangents from a Point?

We know there is exactly one tangent at a point on a circle. But what if the point is not on the circle — what if it is inside, or outside? The answer depends entirely on where the point sits, and there are just three cases.

Take a circle and a point $P$, and try to draw tangents from $P$.

The Three Cases

Case 1 — $P$ inside the circle: 0 tangents. Every line drawn through a point inside the circle cuts the circle at two points, so it is always a secant. You cannot draw a tangent through an interior point.

Case 2 — $P$ on the circle: exactly 1 tangent. This is the case from the last section — the single tangent perpendicular to the radius at $P$.

Case 3 — $P$ outside the circle: exactly 2 tangents. From an external point you can draw two tangents to the circle, touching it at two different points of contact.

Position of the point	Number of tangents
Inside the circle	0
On the circle	1
Outside the circle	2

Key Point: 0, 1, 2 tangents ↔ point inside, on, outside. This tiny table is worth easy marks.

Length of the Tangent from an External Point

In Case 3, let the two tangents from the external point $P$ touch the circle at $T_1$ and $T_2$. The segment from $P$ to a point of contact, $PT_1$ (or $PT_2$), is called the length of the tangent from $P$ to the circle.

There are two such lengths, $PT_1$ and $PT_2$. Measure them and you will notice something striking — they are always equal. That equality is so important that we prove it as a theorem in the next section.

Key Point: From an external point there are two tangents, and (as Theorem 10.2 will show) the two tangent lengths are equal.

Solved Examples

Example 1: Counting tangents

How many tangents can be drawn to a circle from (a) a point inside it, (b) a point on it, (c) a point outside it?

Solution:

1. Inside: every line through it is a secant, so 0 tangents.
2. On the circle: 1 tangent (perpendicular to the radius there).
3. Outside: 2 tangents.

Final Answer: (a) 0, (b) 1, (c) 2.

Takeaway: The count is fixed by the position of the point.

Example 2: Is a tangent possible?

A point $P$ lies at a distance of 4 cm from the centre of a circle of radius 6 cm. How many tangents can be drawn from $P$?

Solution:

1. Compare the distance to the radius: $4 < 6$, so $P$ is inside the circle.
2. From an interior point, no tangent can be drawn.

Final Answer: 0 tangents.

Takeaway: Compare distance $d$ with radius $r$: $d<r$ inside (0), $d=r$ on (1), $d>r$ outside (2).

Example 3: Two tangents

The two tangents from an external point $P$ touch a circle at $A$ and $B$. If $PA = 7$ cm, what is $PB$?

Solution:

1. The two tangents from the same external point are equal in length (Theorem 10.2).
2. So $PB = PA = 7$ cm.

Final Answer: $PB = 7$ cm.

Takeaway: Equal tangents from an external point — a fact used in almost every circumscribed-figure problem.