Tangent of a Circle

What is a Tangent of a Circle

The Latin word ‘tangent’ means, ‘to touch’. The tangent of a circle is defined as a straight line that touches the circle at a single point. The point where the tangent touches the circle is called the ‘point of tangency’ or the ‘point of contact’. At the point of tangency, a tangent is perpendicular to the radius of the circle.

Shown below is a tangent to the given circle.

Let’s learn some of the common rules of the tangent line to a circle.

Rules of Tangent Lines to a Circle

1. Always touches the circle at only one point
2. Never crosses a circle, meaning it does not pass through the circle
3. It is perpendicular to the radius of the circle
4. The length of tangents from a common external point is equal

Conditions of Tangency

A line or a line segment is considered a tangent only when it touches a curve at a single point or else it is simply a line or a line segment. Thus, based on the point of tangency and its location with respect to the circle, there can be three possible conditions for the tangent as given below:

When the Point Lies Inside the Circle

If ‘P’ is a point inside the circle, all the lines through point ‘P’ intersect the circle at two points. Thus, no tangent can be drawn to a circle, which passes through a point lying inside the circle. 

When the Point Lies On the Circle

If ‘P’ is a point on the circle, then from the given figure, we can say that there is only one tangent possible through a given point on the circle.

When the Point Lies Outside the Circle

If ‘P’ is a point outside the circle, then from the given figure, we can say that there are exactly two tangents to circle from a point which lies outside the circle.

Formula

How to Find the Tangent of a Circle

The formula to calculate the tangent of a circle is derived below:

Suppose a point ‘T’ lies outside the circle. From that point, we will draw two tangents to the circle. In the given figure, if ‘RT’ is the secant and ‘QT’ is the tangent of the circle, then the relation between the secant and the tangent is given as:

RT/QT = QT/PT

By cross multiplying, we get,

QT² = RT × PT

Tangent Theorems

Tangent Radius Theorem

Two-Tangents Theorem

Let’s work out a few example problems involving the tangent of a circle.

Solved Problems

General Equation of Tangent to a Circle

The standard form of the equation of a circle with center at (h, k), radius r, and a point L (x, y) on the circumference of the circle is given by:

r² = (x-h)² + (y-k)²

Let us learn the steps to find the equation of the tangent to a circle using the above formula.

Steps To Determine the Equation of a Tangent

In order to determine the equation of a tangent, follow the following steps:

1) Determine the equation of the circle and write it in the form:

r² = (x-h)² + (y-k)²

2)  From the equation, determine the coordinates of the center of the circle (h, k)

3) Determine the slope of the radius of the circle by using the formula:

m_OL = y₂ – y₁/x₂ – x₁

4) The radius OP being perpendicular to the tangent PQ, it is written as

m_PQ =-1/m_OL

5) Write the slope-intercept form of the equation of a circle substituting the value of m_PQ and the coordinates of P, making ‘y’ the subject of the equation:

y – y₁ = m(x – x₁)

Solved Problems

Let’s work out a problem to clear your concept better.