Tangents to Circles (Cambridge International IGCSE Further Maths)

Tangents to circles

What is a tangent to a circle?

• A tangent (to a circle) is a line that touches the circle at a single point but does not cut across the circle
• A tangent to a circle is perpendicular to the radius of the circle at the point of intersection

How can I find the equation of the tangent line to a circle at a given point?

• STEP 1 Find the gradient of the radius OP

• STEP 2 Find the gradient of the tangent As they are perpendicular, if the gradient of the radius is $m_{r}$, then, the gradient of the tangent is $m_{t}=-\frac{1}{m_{r}}$ (Negative reciprocal) Alternatively,

• STEP 3 The equation of the tangent is the equation of the line with that gradient that goes through point P (x₂, y₂) Write down the equation using y−y_ 2=m_ t (x−x_ 2)where $m_{t}$is the gradient

Exam Hint

• If you understand the formula in Step 2 above

Gradient of tangent $=-\frac{x_{2}-x_{1}}{y_{2}-y_{1}}$

you can find the gradient of the tangent directly without having to find the gradient of the radius first.

Worked Example