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Circles

Equations and graphs of circles

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Circles

Standard Form

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Where:

  • Center: (h,k)(h, k)
  • Radius: rr

Example

Circle with center (3,−2)(3, -2) and radius 55: (x−3)2+(y+2)2=25(x - 3)^2 + (y + 2)^2 = 25

General Form

x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0

To convert to standard form:

  1. Group xx terms and yy terms
  2. Complete the square for both
  3. Identify center and radius

Finding Center and Radius

From: x2+y2−6x+4y−12=0x^2 + y^2 - 6x + 4y - 12 = 0

Step 1: Group and move constant (x2−6x)+(y2+4y)=12(x^2 - 6x) + (y^2 + 4y) = 12

Step 2: Complete the square (x2−6x+9)+(y2+4y+4)=12+9+4(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4

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

Center: (3,−2)(3, -2), Radius: 55

Graphing

  1. Plot the center point
  2. Count rr units in all directions
  3. Sketch the circle

📚 Practice Problems

1Problem 1easy

❓ Question:

Write the equation of a circle with center (0,0)(0, 0) and radius 44.

💡 Show Solution

Use standard form: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Center: (h,k)=(0,0)(h, k) = (0, 0) Radius: r=4r = 4

(x−0)2+(y−0)2=42(x - 0)^2 + (y - 0)^2 = 4^2

Simplify: x2+y2=16x^2 + y^2 = 16

Answer: x2+y2=16x^2 + y^2 = 16

2Problem 2medium

❓ Question:

Find the center and radius of (x+1)2+(y−4)2=36(x + 1)^2 + (y - 4)^2 = 36

💡 Show Solution

Compare to standard form: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Rewrite as: (x−(−1))2+(y−4)2=62(x - (-1))^2 + (y - 4)^2 = 6^2

Center: (−1,4)(-1, 4)

Radius: r=36=6r = \sqrt{36} = 6

Answer: Center (−1,4)(-1, 4), radius 66

3Problem 3hard

❓ Question:

Convert to standard form and find the center and radius: x2+y2+8x−2y+8=0x^2 + y^2 + 8x - 2y + 8 = 0

💡 Show Solution

Step 1: Group variables (x2+8x)+(y2−2y)=−8(x^2 + 8x) + (y^2 - 2y) = -8

Step 2: Complete the square

  • For xx: (8/2)2=16(8/2)^2 = 16
  • For yy: (−2/2)2=1(-2/2)^2 = 1

(x2+8x+16)+(y2−2y+1)=−8+16+1(x^2 + 8x + 16) + (y^2 - 2y + 1) = -8 + 16 + 1

Step 3: Factor and simplify (x+4)2+(y−1)2=9(x + 4)^2 + (y - 1)^2 = 9

Answer:

  • Standard form: (x+4)2+(y−1)2=9(x + 4)^2 + (y - 1)^2 = 9
  • Center: (−4,1)(-4, 1)
  • Radius: 33
Explain using:

⚠️ Common Mistakes: Circles

Avoid these 3 frequent errors

🌍 Real-World Applications: Circles

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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Parabolas (Conic Form)
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📌 Related Topics in Conic Sections

Parabolas (Conic Form)

❓ Frequently Asked Questions

What is Circles?▾
Equations and graphs of circles
How can I study Circles effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Circles study guide free?▾
Yes — all study notes, flashcards, and practice problems for Circles on Study Mondo are free to access. No account is needed.
What course covers Circles?▾
Circles is part of the Algebra 2 course on Study Mondo, specifically in the Conic Sections section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Circles?▾
Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.