Equation of Circles and Tangents for GCSE Mathematics

Introduction

In GCSE Mathematics, understanding the equation of circles and tangents is crucial for solving geometry problems accurately. This guide will provide a comprehensive overview of these concepts, including key formulas, worked examples, and exam tips.

Key Concepts

Equation of a Circle:
$$x^2 + y^2 = r^2$$ where (x, y) are the coordinates of any point on the circle and r is the radius.
Tangent Line:
A line that touches the circle at exactly one point.
Point of Tangency:
The point where the tangent line meets the circle.

Equation of a Tangent Line

The equation of a tangent to a circle with center (h, k) and radius r that touches the circle at point (x₁, y₁) is given by:

$$y - y₁ = m(x - x₁)$$

where m = the gradient of the normal at the point of tangency.

Worked Examples

Example 1: Find the equation of the circle with center (2, 3) and radius 4.
Equation: $(x 2)^2 + (y + 3)^2 = 4^2$
Example 2: Find the equation of the tangent to the circle $x^2 + y^2 = 16$ at the point (3, 4).
Gradient of the normal: $m = \frac{3}{4}$
Equation: $y 4 = \frac{3}{4}(x 3)$

Common Mistakes

Forgetting to square the coordinates when substituting into the equation of a circle.
Confusing the center of the circle with the point of tangency when writing the equation of a tangent line.
Assuming that the gradient of the tangent line is the same as the radius of the circle.

Conclusion

Mastering the equation of circles and tangents is essential for success in GCSE Mathematics. By understanding the key concepts and formulas, you can confidently solve geometry problems and prepare effectively for your экзамены. Remember to practice regularly, avoid common mistakes, and utilize our recommended resources for additional support.

FAQ

1. How do I find the point of tangency on a circle?
Draw the normal to the tangent line at the point of tangency and find its intersection with the circle.
2. Can a circle have multiple tangent lines at the same point?
No, a circle can only have one tangent line at any given point.