📌 What is a Circle?

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.

Standard Form (Center-Radius Form):

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

General Form (Expanded Form):

\( x^2 + y^2 + Dx + Ey + F = 0 \)

Where \( D, E, \) and \( F \) are constants. This form requires conversion to standard form to find center and radius.

Special Case - Circle at Origin:

\( x^2 + y^2 = r^2 \)

When the center is at the origin \( (0, 0) \), the equation simplifies significantly.

From Standard Form:

If the equation is \( (x - h)^2 + (y - k)^2 = r^2 \):

Center = \( (h, k) \)

⚠️ Important: Watch the signs! \( (x - h) \) means center x-coordinate is \( +h \), and \( (x + h) \) means center x-coordinate is \( -h \)

📝 Examples - Finding Center:

Example 1: \( (x - 3)^2 + (y - 5)^2 = 16 \)

Center: \( (3, 5) \)

Example 2: \( (x + 2)^2 + (y - 4)^2 = 25 \)

Rewrite: \( (x - (-2))^2 + (y - 4)^2 = 25 \)
Center: \( (-2, 4) \)

Example 3: \( x^2 + (y + 6)^2 = 9 \)

Rewrite: \( (x - 0)^2 + (y - (-6))^2 = 9 \)
Center: \( (0, -6) \)

From Standard Form:

If the equation is \( (x - h)^2 + (y - k)^2 = r^2 \):

📝 Examples - Finding Radius and Diameter:

Example 1: \( (x - 1)^2 + (y + 3)^2 = 49 \)

\( r^2 = 49 \), so \( r = \sqrt{49} = 7 \)
Radius: 7
Diameter: \( 2(7) = 14 \)

Example 2: \( (x + 5)^2 + y^2 = 100 \)

\( r^2 = 100 \), so \( r = \sqrt{100} = 10 \)
Radius: 10
Diameter: 20

Example 3: \( x^2 + y^2 = 20 \)

\( r = \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \approx 4.47 \)
Radius: \( 2\sqrt{5} \)
Diameter: \( 4\sqrt{5} \)

Steps:

1. Identify the center \( (h, k) \) from the graph
2. Find the radius by counting units from center to any point on the circle
3. Square the radius to get \( r^2 \)
4. Write the equation: \( (x - h)^2 + (y - k)^2 = r^2 \)

📝 Example - From Graph:

A circle has center at \( (2, -3) \) and passes through \( (2, 1) \). Write the equation.

Step 1: Center: \( (2, -3) \)
Step 2: Find radius using distance formula:
\( r = \sqrt{(2-2)^2 + (1-(-3))^2} = \sqrt{0 + 16} = 4 \)
Step 3: \( r^2 = 16 \)
Equation: \( (x - 2)^2 + (y + 3)^2 = 16 \)

Given Center and Radius:

Simply substitute the values into the standard form equation.

Given Endpoints of Diameter:

1. Find the center using midpoint formula: \( \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)
2. Find the radius using distance from center to either endpoint
3. Write the equation using center and radius

📝 Example - From Diameter Endpoints:

A circle has diameter with endpoints \( (2, 4) \) and \( (8, 12) \). Find the equation.

Step 1: Find center (midpoint):
\( h = \frac{2+8}{2} = 5, \quad k = \frac{4+12}{2} = 8 \)
Center: \( (5, 8) \)

Step 2: Find radius (distance from center to endpoint):
\( r = \sqrt{(5-2)^2 + (8-4)^2} = \sqrt{9+16} = \sqrt{25} = 5 \)

Equation: \( (x - 5)^2 + (y - 8)^2 = 25 \)

Method: Completing the Square

To convert \( x^2 + y^2 + Dx + Ey + F = 0 \) to standard form:

1. Move the constant to the right side
2. Group x-terms and y-terms separately
3. Complete the square for x: add \( \left(\frac{D}{2}\right)^2 \) to both sides
4. Complete the square for y: add \( \left(\frac{E}{2}\right)^2 \) to both sides
5. Factor perfect square trinomials
6. Simplify to get standard form

📝 Example - Completing the Square:

Convert \( x^2 + y^2 - 6x + 4y - 12 = 0 \) to standard form

Step 1: Move constant:
\( x^2 + y^2 - 6x + 4y = 12 \)

Step 2: Group terms:
\( (x^2 - 6x) + (y^2 + 4y) = 12 \)

Step 3: Complete the square for x:
Half of -6 is -3, squared is 9
\( (x^2 - 6x + 9) + (y^2 + 4y) = 12 + 9 \)

Step 4: Complete the square for y:
Half of 4 is 2, squared is 4
\( (x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4 \)

Step 5: Factor:
\( (x - 3)^2 + (y + 2)^2 = 25 \)

Center: \( (3, -2) \), Radius: 5

📝 Example 2 - More Practice:

Convert \( x^2 + y^2 + 8x - 2y + 8 = 0 \)

Move constant: \( x^2 + y^2 + 8x - 2y = -8 \)
Group: \( (x^2 + 8x) + (y^2 - 2y) = -8 \)
For x: \( \left(\frac{8}{2}\right)^2 = 16 \)
For y: \( \left(\frac{-2}{2}\right)^2 = 1 \)
\( (x^2 + 8x + 16) + (y^2 - 2y + 1) = -8 + 16 + 1 \)
\( (x + 4)^2 + (y - 1)^2 = 9 \)

Center: \( (-4, 1) \), Radius: 3

Formulas from \( x^2 + y^2 + Dx + Ey + F = 0 \):

📝 Example - Direct from General Form:

Find center and radius of \( x^2 + y^2 - 10x + 6y + 9 = 0 \)

\( D = -10, E = 6, F = 9 \)

Center:
\( h = -\frac{-10}{2} = 5 \)
\( k = -\frac{6}{2} = -3 \)
Center: \( (5, -3) \)

Radius:
\( r = \sqrt{\left(\frac{-10}{2}\right)^2 + \left(\frac{6}{2}\right)^2 - 9} \)
\( = \sqrt{25 + 9 - 9} = \sqrt{25} = 5 \)

Step-by-Step Graphing:

1. Convert to standard form if necessary
2. Identify the center \( (h, k) \) and plot it
3. Identify the radius \( r \)
4. Plot four key points by moving \( r \) units:
   • Up from center: \( (h, k + r) \)
   • Down from center: \( (h, k - r) \)
   • Right from center: \( (h + r, k) \)
   • Left from center: \( (h - r, k) \)
5. Draw a smooth circle through these four points

📝 Example - Complete Graphing:

Graph \( (x - 2)^2 + (y + 1)^2 = 16 \)

Step 1: Identify center and radius
Center: \( (2, -1) \)
Radius: \( r = \sqrt{16} = 4 \)

Step 2: Plot center at \( (2, -1) \)

Step 3: Plot four key points:
Up: \( (2, -1 + 4) = (2, 3) \)
Down: \( (2, -1 - 4) = (2, -5) \)
Right: \( (2 + 4, -1) = (6, -1) \)
Left: \( (2 - 4, -1) = (-2, -1) \)

Step 4: Draw smooth circle through these points

📝 Example 2 - From General Form:

Graph \( x^2 + y^2 - 4x + 6y - 3 = 0 \)

Step 1: Convert to standard form:
\( (x^2 - 4x) + (y^2 + 6y) = 3 \)
\( (x^2 - 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9 \)
\( (x - 2)^2 + (y + 3)^2 = 16 \)

Step 2: Center: \( (2, -3) \), Radius: 4

Step 3: Key points:
\( (2, 1) \), \( (2, -7) \), \( (6, -3) \), \( (-2, -3) \)

Step 4: Draw the circle

⚡ Quick Summary

• Standard form makes it easy to identify center and radius
• Watch signs carefully: \( (x - h) \) means center at \( +h \)
• Use completing the square to convert general to standard form
• All points on a circle are exactly \( r \) units from the center
• To graph: plot center, mark 4 key points, draw smooth curve

📚 Key Formulas Reference