Tangent Perpendicular to Radius

Statement

In coordinate geometry, a tangent to a circle at a point is perpendicular to the radius through that point. Perpendicular lines have slopes (gradients) that multiply to –1. This gives the relation:

\[ m_{\text{tan}} \times m_{\text{radius}} = -1 \]

Here, \(m_{\text{tan}}\) is the gradient of the tangent, and \(m_{\text{radius}}\) is the gradient of the radius. If you know one gradient, you can calculate the other using this rule.

Why it’s true

• The tangent is perpendicular to the radius at the point of contact.
• In coordinate geometry, perpendicular lines have gradients that are negative reciprocals: \(m_1 \cdot m_2 = -1\).
• So, the gradient of the tangent can be found once the gradient of the radius is known, and vice versa.

Recipe (how to use it)

1. Find the coordinates of the point of tangency.
2. Calculate the gradient of the radius from the circle’s centre to that point.
3. Use \(m_{\text{tan}} = -\tfrac{1}{m_{\text{radius}}}\).
4. Form the tangent equation using point-gradient form.

Spotting it

Whenever you see a circle in coordinate geometry problems, the tangent gradient is linked to the radius gradient. This is especially useful when deriving the tangent equation to a circle.

Common pairings

• Equation of a circle: \((x-a)^2+(y-b)^2=r^2\).
• Equation of a line: \(y-y_1=m(x-x_1)\).

Mini examples

1. Given: Centre (0,0), point (3,4). Radius gradient = \(4/3\). Answer: Tangent gradient = \(-3/4\).
2. Given: Centre (1,2), point (5,6). Radius gradient = \(\tfrac{6-2}{5-1}=1\). Answer: Tangent gradient = -1.

Pitfalls

• Forgetting to use negative reciprocal (students often use just negative).
• Not simplifying gradients properly.
• Mixing up the point of tangency with other points on the circle.

Exam strategy

• Always find the radius gradient first — it’s straightforward from the centre to the tangent point.
• Write gradients as fractions to make the reciprocal step clearer.
• Use point–slope form for tangent equations.

Summary

The tangent to a circle is perpendicular to the radius at the contact point. In coordinate geometry, this translates to \(m_{\text{tan}} \cdot m_{\text{radius}} = -1\). It’s a key tool for finding tangent equations and solving circle problems.