Lesson Plan

• Recall the standard and general forms of a circle and identify its centre and radius.
• Derive the tangency condition for a line and a circle using the distance formula.
• Find the equation of the tangent at a given point on a circle.
• Determine the equations of tangents drawn from an external point by using the discriminant or point‑form method.
• Apply these techniques to solve exam‑style problems involving tangents to circles.

• Projector and screen
• Whiteboard and markers
• Printed worksheet with practice questions
• Graph paper, ruler and compass
• Scientific calculators
• GeoGebra (or similar) demonstration on the computer

Begin with a short video clip showing how road engineers use tangents to design safe curves, linking the mathematics to a real‑world context. Ask students to recall the equation of a circle and the distance formula from previous lessons. State that by the end of the lesson they will be able to derive and use the tangency condition to find equations of tangents and solve typical exam questions.

1. Do‑now (5'): Quick quiz on standard and general forms of a circle.
2. Mini‑lecture (10'): Derive the tangency condition \((ah+bk+c)^2=r^2(a^2+b^2)\) and illustrate with a simple example.
3. Guided example (15'): Work through the external‑point problem (circle \(x^2+y^2=25\), point \(E(7,1)\)) using the discriminant method, showing each algebraic step.
4. Pair activity (10'): Students use the point‑form formula to find the tangent at a given point on a shifted circle, checking their work with GeoGebra.
5. Whole‑class discussion (5'): Share solutions, highlight common errors, and reinforce key formulas.
6. Exit ticket (5'): Each student writes the equation of a tangent from a new external point (e.g., \(A(8,2)\)) on a blank sheet.

Summarise the five core formulas presented today and emphasise the link between the distance approach and the discriminant approach. Collect exit tickets to gauge understanding, and assign homework: complete the remaining practice questions (items 2‑5) in the worksheet and prepare a short explanation of when each method is most efficient.