Equation of a circle

In this section, we will discuss how to work with the equation of a circle. Specifically, we will cover how to find the centre and radius from the equation of a circle, how to find the equation from the endpoints of a diameter, and how to determine the centre and radius from the general form of the equation \( x^2 + y^2 + ax + by + c = 0 \).

1. Finding the Centre and Radius from the Equation of a Circle

The general equation of a circle is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the centre and \( r \) is the radius of the circle. This equation is in standard form, which makes it easy to identify the centre and the radius.

If the equation of the circle is given in expanded form, such as:

\[ x^2 + y^2 + ax + by + c = 0 \] you can rewrite this equation into the standard form to find the centre and radius. To do this, you need to complete the square for both \( x \) and \( y \).

Here’s the step-by-step process for finding the centre and radius from the expanded equation:

1. Group the \( x \)-terms and \( y \)-terms: Rearrange the equation to group the \( x \)-terms and \( y \)-terms together.
2. Complete the square for the \( x \)-terms and \( y \)-terms: For the \( x \)-terms, take half of the coefficient of \( x \), square it, and add it to both sides. Do the same for the \( y \)-terms.
3. Rewrite the equation: The equation will now be in the form:

2. Finding the Equation of a Circle from the Endpoints of a Diameter

If you are given the endpoints of the diameter of the circle, you can find the equation of the circle by following these steps:

1. Find the centre: The centre of the circle is the midpoint of the diameter. If the endpoints of the diameter are \( (x_1, y_1) \) and \( (x_2, y_2) \), the centre \( (h, k) \) is:
2. Find the radius: The radius is half the length of the diameter. The length of the diameter is the distance between the two endpoints, which is given by:
3. Write the equation of the circle: Once you have the centre \( (h, k) \) and the radius \( r \), the equation of the circle is:

3. Finding the Centre and Radius from the General Equation \( x^2 + y^2 + ax + by + c = 0 \)

To find the centre and radius of the circle from the general equation \( x^2 + y^2 + ax + by + c = 0 \), you follow the process of completing the square as mentioned earlier. Let’s go through this step by step.

1. Group the \( x \)-terms and \( y \)-terms: Rearrange the equation to group the \( x \)-terms and \( y \)-terms together:
2. Complete the square for both the \( x \)-terms and \( y \)-terms: Add \( \left( \frac{a}{2} \right)^2 \) to both sides for the \( x \)-terms, and add \( \left( \frac{b}{2} \right)^2 \) to both sides for the \( y \)-terms.
3. Find the centre and radius: The centre of the circle is \( \left( -\frac{a}{2}, -\frac{b}{2} \right) \), and the radius is given by:

4. Summary:

• The equation of a circle in standard form is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the centre and \( r \) is the radius.
• To find the equation of a circle from the endpoints of the diameter, find the centre by calculating the midpoint, and find the radius by calculating half the distance between the endpoints.
• To find the centre and radius from the general equation \( x^2 + y^2 + ax + by + c = 0 \), complete the square to rewrite the equation in standard form.

These techniques are essential for understanding and working with circles in coordinate geometry, allowing you to find key properties such as the centre, radius, and equation of the circle.