How many tangents can be drawn from an external point to a circle and formula fo

Tangents to a Circle from an External Point Learn how to find the number of tangents and their length from an external point to a circle using coordinate geometry.

Concept Overview

This question tests the fundamental understanding of tangents to a circle, specifically from a point outside the circle. It involves visualizing the geometric configuration and applying the distance formula along with the Pythagorean theorem to derive the formula for the length of a tangent segment. The core idea is that from any external point, exactly two tangents can be drawn to a given circle.

Step 1: Understanding the Geometry Consider a circle with center $C$ and radius $r$. Let $P$ be an external point. A tangent from $P$ to the circle touches the circle at a point, say $T$. The radius $CT$ is perpendicular to the tangent line $PT$ at the point of tangency $T$. This forms a right-angled triangle $\triangle CTP$, with the right angle at $T$.

Step 2: Determining the Number of Tangents If a point $P$ is outside the circle, we can draw two distinct lines from $P$ that are tangent to the circle. Let the distance of the point $P$ from the center $C$ be $d$. If $d > r$, the point $P$ is outside the circle, and two tangents can be drawn. If $d = r$, the point $P$ lies on the circle, and only one tangent can be drawn. If $d < r$, the point $P$ is inside the circle, and no tangents can be drawn.

Step 3: Deriving the Length of a Tangent In the right-angled triangle $\triangle CTP$, the hypotenuse is the distance from the center to the external point, $CP = d$. The sides are the radius $CT = r$ and the length of the tangent segment $PT$. By the Pythagorean theorem, we have:

Substituting the known values:

Rearranging the formula to find the length of the tangent $PT$:

Therefore, the length of the tangent segment from an external point $P$ to a circle is:

This formula is valid only when $d > r$, ensuring that $d^2 - r^2$ is positive.

Step 4: Example Calculation Suppose a circle has the equation $(x-2)^2 + (y-3)^2 = 16$. This means the center $C$ is at $(2, 3)$ and the radius $r = \sqrt{16} = 4$. Let the external point $P$ be $(6, 6)$. First, calculate the distance $d$ between the center $C(2, 3)$ and the point $P(6, 6)$ using the distance formula:

Since $d = 5$ and $r = 4$, we have $d > r$, so the point $P$ is indeed external to the circle. Now, calculate the length of the tangent using the formula $PT = \sqrt{d^2 - r^2}$:

Thus, the length of the tangent from point $(6, 6)$ to the given circle is 3 units.

Key Takeaways:

• From an external point to a circle, exactly two tangents can be drawn.
• The radius drawn to the point of tangency is perpendicular to the tangent line.
• The length of the tangent segment from an external point $P$ to a circle with center $C$ and radius $r$ is given by $\sqrt{d^2 - r^2}$, where $d$ is the distance $CP$.
• This formula is applicable only when the point is external to the circle ($d > r$).

Answer: The length of the tangent from an external point to a circle is $\sqrt{d^2 - r^2}$, where $d$ is the distance from the external point to the center of the circle and $r$ is the radius of the circle. Exactly two tangents can be drawn from an external point.