Tangents (Leaving Cert Mathematics): Revision Notes

Tangents

What is a Tangent to a Circle?

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact.

Key Properties of Tangents

1. Perpendicularity: A tangent is perpendicular to the radius at the point of contact.
2. Uniqueness: There is only one tangent to a circle at a given point on its circumference.
3. Common Tangents: Two circles may share a common tangent, either internally or externally.

Tangents from an External Point

From a point outside a circle, two tangents can be drawn to the circle. The lengths of these tangents are equal.

Finding the Equation of a Tangent

Case 1: Tangent at a Known Point $(x_1, y_1)$

For a circle with equation $(x - h)^2 + (y - k)^2 = r^2$, the equation of the tangent at $(x_1, y_1)$ is:

$$(x_1 - h)(x - h) + (y_1 - k)(y - k) = r^2$$

Case 2: Tangent from an External Point $(x_1, y_1)$

For a circle with equation $x^2 + y^2 = r^2$ (centred at the origin), the tangents from an external point $(x_1, y_1)$ can be found using the following steps:

1. Find the distance from the point $(x_1, y_1)$ to the centre of the circle.
2. Use the condition that the tangent is perpendicular to the radius at the point of contact.
3. Form simultaneous equations and solve.

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Worked Examples

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Summary

• Definition: A tangent is a line that touches the circle at one point and is perpendicular to the radius at that point.
• Equation of Tangent: Use the point of contact or external points to derive the equation.
• Key Properties:
  1. Perpendicular to the radius.
  2. Unique for each point of contact.
  3. Equal tangents from an external point.
• Practice deriving tangent equations to understand the relationships between lines and circles.