8.3 Circle Tangents

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Understand what a tangent to a circle is
  • Show that a given line is a tangent to a circle
  • Find the equation of a tangent to a circle at a given point

📖 Main Content

https://cambridgeassistant.com

1. What Is a Tangent?

A tangent is a straight line that touches a circle at exactly one point. It does not cut through the circle — it just grazes it at a single spot. That single point where the tangent meets the circle is called the point of tangency (or point of contact).

Think of a ball sitting on a flat floor — the floor only touches the ball at one point. That floor is like a tangent to the ball.

2. The Key Geometric Property of a Tangent

This is the most important rule you need to remember:

A tangent to a circle is always perpendicular (at 90°) to the radius drawn to the point of tangency.

In other words, if you draw a line from the centre of the circle to the point where the tangent touches the circle, that line (the radius) and the tangent will always form a right angle.

Why does this matter? Because it gives you a way to find the equation of a tangent without using calculus. If you know the gradient (steepness) of the radius, you can immediately find the gradient of the tangent — since perpendicular lines have gradients that multiply to give −1.

Rule for perpendicular gradients: If two lines are perpendicular, then: m1×m2=−1m_1 \times m_2 = -1 So if the radius has gradient m1m_1, the tangent has gradient m2=−1m1m_2 = \dfrac{-1}{m_1}

3. How to Find the Equation of a Tangent at a Given Point

Follow these steps every time:

Step 1: Identify the centre of the circle from its equation. (Rewrite the circle equation in the form (x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2 if needed — the centre is (a,b)(a, b).)

Step 2: Find the gradient of the radius that goes from the centre to the given point of tangency. Use the gradient formula: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Step 3: Find the gradient of the tangent using the perpendicular rule:

mtangent=−1mradiusm_{\text{tangent}} = \frac{-1}{m_{\text{radius}}}

Step 4: Write the equation of the tangent using the point-slope form: y−y1=m(x−x1)y - y_1 = m(x - x_1) where (x1,y1)(x_1, y_1) is the point of tangency.

Sign in to view full notes