Integrated Math 2: Equations of Circles: Step-by-Step Instructions

Understanding Equations of Circles ⭕

In Integrated Math 2, understanding the equation of a circle is fundamental. A circle's equation allows us to define its position and size on a coordinate plane. Let's break down the key concepts and provide a step-by-step guide.

The Standard Form 📝

The standard form of the equation of a circle 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.

Steps to Write the Equation of a Circle ✍️

1. Identify the Center: Determine the coordinates $(h, k)$ of the center of the circle.
2. Find the Radius: Determine the radius $r$ of the circle. This is the distance from the center to any point on the circle.
3. Plug in the Values: Substitute the values of $h$, $k$, and $r$ into the standard form equation.
4. Simplify: Simplify the equation if necessary.

Example 1: Writing the Equation from Center and Radius 💻

Suppose a circle has a center at $(2, -3)$ and a radius of $5$. Write the equation of the circle.

1. Identify the Center: $(h, k) = (2, -3)$
2. Find the Radius: $r = 5$
3. Plug in the Values: $(x - 2)^2 + (y - (-3))^2 = 5^2$
4. Simplify: $(x - 2)^2 + (y + 3)^2 = 25$

Therefore, the equation of the circle is $(x - 2)^2 + (y + 3)^2 = 25$.

Example 2: Writing the Equation from a Graph 📈

Suppose you have a circle graphed on the coordinate plane. The center is at $(-1, 4)$ and the circle passes through the point $(2, 8)$. Write the equation of the circle.

1. Identify the Center: $(h, k) = (-1, 4)$
2. Find the Radius: Use the distance formula to find the radius $r$ between the center $(-1, 4)$ and the point $(2, 8)$.

The distance formula is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

So, $r = \sqrt{(2 - (-1))^2 + (8 - 4)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

1. Plug in the Values: $(x - (-1))^2 + (y - 4)^2 = 5^2$
2. Simplify: $(x + 1)^2 + (y - 4)^2 = 25$

Therefore, the equation of the circle is $(x + 1)^2 + (y - 4)^2 = 25$.

General Form to Standard Form ⚙️

Sometimes, the equation of a circle is given in the general form:

$x^2 + y^2 + Ax + By + C = 0$

To convert it to standard form, you need to complete the square for both $x$ and $y$ terms.

Example 3: Converting from General to Standard Form 💡

Convert the following equation to standard form: $x^2 + y^2 - 4x + 6y - 23 = 0$

1. Group $x$ and $y$ terms: $(x^2 - 4x) + (y^2 + 6y) = 23$
2. Complete the square for $x$: $(x^2 - 4x + 4) + (y^2 + 6y) = 23 + 4$
3. Complete the square for $y$: $(x^2 - 4x + 4) + (y^2 + 6y + 9) = 23 + 4 + 9$
4. Factor and Simplify: $(x - 2)^2 + (y + 3)^2 = 36$

Now, the equation is in standard form: $(x - 2)^2 + (y + 3)^2 = 36$. The center is $(2, -3)$ and the radius is $6$.

Key Takeaways 🔑

• The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$.
• $(h, k)$ represents the center of the circle.
• $r$ represents the radius of the circle.
• Completing the square helps convert the general form to standard form.