Diving Into Circles and Tangents

What is the Equation of a Circle?

The equation of a circle describes all the points that are the same distance (called the radius) from a fixed center point. It’s like drawing a perfect circle around a center.

The general equation of a circle is:

$$x ^2 + y^2 = r^2$$

Where $r$ is the radius of the circle.

 

Inside, On or Outside?

When given a co-ordinate you can work out whether the point lies on, inside or outside the circle using the equation. Lets say we have the co-ordinate $(p, q)$ we can substitute it in to the equation:

• If $p^2 + q^2 = r^2$ then it lies on the circle
• If $p^2 + q^2 \gt r^2$ then it is outside the circle
• If $p^2 + q^2 \lt r^2$ then it is inside the circle

 

Example

Determine if the Point $(4,0)$ Lies on the Circle $x^2 + y^2 = 16$:

Solution:

1. Substitute $(x,y) = (4,0)$ into the equation: $$x^2 + y^2 = 16 \\ \text{Substitute} \ x = 4 \text{ and} \ y = 0: 4^2 + 0^2 = 16 \\  16 = 16$$
2. The point satisfies the equation, so $(4,0)$ lies on the circle

 

 

 

 

Tangents to a Circle

What is a Tangent to a Circle?

A tangent is a straight line that touches a circle at exactly one point. This point is called the point of tangency. The tangent is always perpendicular to the radius at the point of tangency.

 

Key Idea

For a circle with the equation: $$x^2 + y^2 = r^2$$

• The radius connects the center of the circle (at the origin $(0, 0)$ to the point of tangency.
• The gradient of the tangent is the negative reciprocal of the gradient of the radius.

 

Steps to Find the Equation of a Tangent

1. Find the Gradient of the Radius:

   • Calculate the gradient of the line from the center $(0, 0)$ to the given point of tangency $x_1, y_1)$: $$\text{Gradient of radius} = \frac{y_1 - 0}{x_1 - 0} = \frac{y_1}{x_1}$$

2. Find the Gradient of the Tangent:

   • The tangent is perpendicular to the radius, so its gradient is: $$\text{Gradient of tangent} = -\frac{x_1}{y_1}$$

3. Use the Point-Gradient Formula:

   • The equation of a line is given by: $$y - y_1 = m(x - x_1)$$
   • Here, $m = -\frac{x_1}{y_1}$

4. Simplify the Equation:

   • Expand and rearrange to find the tangent's equation in standard form.

 

Examples

Example 1: Find the Equation of the Tangent to the Circle $x^2 + y^2 = 25$ at the point $(3, 4)$:

1. Find the Gradient of the Radius: The radius connects the center $(0, 0)$ to $(3, 4)$ so: $$\text{Gradient of radius} = \frac{4}{3}$$
2. Find the Gradient of the Tangent: The tangent is perpendicular to the radius, so its gradient is: $$\text{Gradient of tangent} = -\frac{3}{4}$$
3. Write the Equation of the Tangent: Using the point-gradient formula with $(x_1, y_1) = (3,4)$ and $m = - \frac{3}{4}$: $$y - 4 = -\frac{3}{4}(x - 3)$$
4. Simplify: Expand the brackets: $$\text{Expand the brackets: } \ y - 4 = -\frac{3}{4}x + \frac{9}{4} \\ \text{Add} 4 \ (or 16/4) \ \text{to both sides: } \ y = -\frac{3}{4}x + \frac{25}{4} \\ \text{Final equation: } \  y = -\frac{3}{4}x + \frac{25}{4}$$