Equations of Circles

In coordinate geometry, a circle is defined as the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center). There are two main ways to write the equation of a circle: Standard Form and General Form.

The Standard Form

The standard equation of a circle makes it very easy to identify the circle's center and radius. The formula is:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

• $(h, k)$ is the center of the circle.
• $r$ is the radius of the circle.
• $(x, y)$ represents any point on the circle.

Example 1: Writing the Equation

Problem: Write the equation of a circle with center $(3, -2)$ and radius $5$.

Solution:

1. Identify the given values: $h = 3$, $k = -2$, and $r = 5$.
2. Substitute these into the standard form equation: $(x - 3)^2 + (y - (-2))^2 = 5^2$
3. Simplify the equation: $(x - 3)^2 + (y + 2)^2 = 25$

The General Form

If you expand the standard form equation, you get the general form of a circle:

$x^2 + y^2 + Dx + Ey + F = 0$

Where $D$, $E$, and $F$ are constants.

To find the center and radius from the general form, you need to convert it back to standard form. This is done using an algebraic technique called completing the square.

Example 2: Converting to Standard Form

Problem: Convert $x^2 + y^2 - 6x + 4y - 12 = 0$ to standard form and find the center and radius.

Solution:

1. Group the $x$ terms and $y$ terms together, and move the constant to the other side of the equation: $(x^2 - 6x) + (y^2 + 4y) = 12$

2. Complete the square for both $x$ and $y$:

   • For $x$: Take half of $-6$ (which is $-3$) and square it to get $9$.
   • For $y$: Take half of $4$ (which is $2$) and square it to get $4$.
   • Add these numbers inside the parentheses, and add them to the right side of the equation as well to keep it balanced: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$

3. Factor the perfect square trinomials and simplify the right side: $(x - 3)^2 + (y + 2)^2 = 25$

4. Identify the center and radius: Comparing this to $(x - h)^2 + (y - k)^2 = r^2$, we can see:

   • Center $(h, k) = (3, -2)$
   • Radius $r = \sqrt{25} = 5$