Solved Examples: Converting General Form Circle Equations to Standard Form

📚 Quick Study Guide

• 🔍 The general form of a circle equation is: $Ax^2 + Ay^2 + Dx + Ey + F = 0$. Notice that the coefficients of $x^2$ and $y^2$ are the same.
• 📝 The standard form of a circle equation is: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.
• 💡 To convert from general to standard form, you'll need to complete the square for both $x$ and $y$.
• ➕ Remember to add the same values to both sides of the equation when completing the square to maintain balance.
• ➗ If $A$ is not equal to 1, divide the entire equation by $A$ before completing the square.
• ✏️ The steps are: group $x$ and $y$ terms, complete the square for $x$, complete the square for $y$, and simplify to get the standard form.

🧪 Practice Quiz

1. Question 1: What is the standard form of the circle equation $x^2 + y^2 - 4x + 6y - 3 = 0$?
   1. (x - 2)^2 + (y + 3)^2 = 16
   2. (x + 2)^2 + (y - 3)^2 = 16
   3. (x - 2)^2 + (y + 3)^2 = 4
   4. (x + 2)^2 + (y - 3)^2 = 4
2. Question 2: What is the center of the circle represented by the equation $(x + 5)^2 + (y - 2)^2 = 9$?
   1. (5, -2)
   2. (-5, 2)
   3. (5, 2)
   4. (-5, -2)
3. Question 3: What is the radius of the circle represented by the equation $(x - 1)^2 + (y + 4)^2 = 25$?
   1. 5
   2. 25
   3. 625
   4. 12.5
4. Question 4: Convert the equation $x^2 + y^2 + 2x - 8y + 8 = 0$ to standard form.
   1. (x + 1)^2 + (y - 4)^2 = 9
   2. (x - 1)^2 + (y + 4)^2 = 9
   3. (x + 1)^2 + (y - 4)^2 = 3
   4. (x - 1)^2 + (y + 4)^2 = 3
5. Question 5: What is the standard form of $x^2 + y^2 - 6x + 4y - 12 = 0$?
   1. (x - 3)^2 + (y + 2)^2 = 25
   2. (x + 3)^2 + (y - 2)^2 = 25
   3. (x - 3)^2 + (y + 2)^2 = 5
   4. (x + 3)^2 + (y - 2)^2 = 5
6. Question 6: Which of the following equations represents a circle centered at (0, 0) with a radius of 7?
   1. x^2 + y^2 = 49
   2. x^2 + y^2 = 7
   3. (x - 7)^2 + (y - 7)^2 = 0
   4. (x + 7)^2 + (y + 7)^2 = 0
7. Question 7: What is the radius of the circle defined by the equation $(x-3)^2 + (y+1)^2 = 16$?
   1. 16
   2. 4
   3. 8
   4. 256