Solved examples: Converting ellipse equations from general to standard form

📚 Quick Study Guide

• 🔍 The general form of an ellipse equation is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $B = 0$ and $A$ and $C$ have the same sign.
• 📝 The standard form of an ellipse equation centered at $(h, k)$ is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (horizontal major axis) or $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ (vertical major axis), where $a > b$.
• 💡 To convert from general to standard form, complete the square for both $x$ and $y$ terms.
• ➕ Identify the center $(h, k)$ by looking at the values being subtracted from $x$ and $y$ in the standard form.
• 📏 Determine the lengths of the major and minor axes based on the values of $a$ and $b$. The major axis has length $2a$, and the minor axis has length $2b$.

Practice Quiz

1. Question 1: Convert the following equation to standard form: $4x^2 + 9y^2 - 16x + 18y - 11 = 0$
   1. $\frac{(x-2)^2}{9} + \frac{(y+1)^2}{4} = 1$
   2. $\frac{(x+2)^2}{9} + \frac{(y-1)^2}{4} = 1$
   3. $\frac{(x-2)^2}{4} + \frac{(y+1)^2}{9} = 1$
   4. $\frac{(x+2)^2}{4} + \frac{(y-1)^2}{9} = 1$
2. Question 2: Convert the following equation to standard form: $25x^2 + 4y^2 + 150x - 16y + 141 = 0$
   1. $\frac{(x+3)^2}{4} + \frac{(y-2)^2}{25} = 1$
   2. $\frac{(x-3)^2}{4} + \frac{(y+2)^2}{25} = 1$
   3. $\frac{(x+3)^2}{25} + \frac{(y-2)^2}{4} = 1$
   4. $\frac{(x-3)^2}{25} + \frac{(y+2)^2}{4} = 1$
3. Question 3: Convert the following equation to standard form: $16x^2 + y^2 - 96x + 8y + 144 = 0$
   1. $\frac{(x-3)^2}{1} + \frac{(y+4)^2}{16} = 1$
   2. $\frac{(x+3)^2}{1} + \frac{(y-4)^2}{16} = 1$
   3. $\frac{(x-3)^2}{16} + \frac{(y+4)^2}{1} = 1$
   4. $\frac{(x+3)^2}{16} + \frac{(y-4)^2}{1} = 1$
4. Question 4: What is the center of the ellipse given by the equation: $9(x-1)^2 + 4(y+2)^2 = 36$?
   1. (1, 2)
   2. (-1, -2)
   3. (1, -2)
   4. (-1, 2)
5. Question 5: What are the lengths of the major and minor axes of the ellipse given by $\frac{(x+5)^2}{16} + \frac{(y-3)^2}{9} = 1$?
   1. Major axis: 3, Minor axis: 4
   2. Major axis: 8, Minor axis: 6
   3. Major axis: 4, Minor axis: 3
   4. Major axis: 16, Minor axis: 9
6. Question 6: Convert the following equation to standard form: $x^2 + 4y^2 + 6x - 8y + 9 = 0$
   1. $\frac{(x+3)^2}{4} + (y-1)^2 = 1$
   2. $\frac{(x-3)^2}{4} + (y+1)^2 = 1$
   3. $\frac{(x+3)^2}{4} + \frac{(y-1)^2}{1} = 1$
   4. $\frac{(x-3)^2}{4} + \frac{(y+1)^2}{1} = 1$
7. Question 7: Which of the following equations represents an ellipse?
   1. $x^2 - y^2 = 1$
   2. $x^2 + y = 1$
   3. $x^2 + y^2 = 1$
   4. $4x^2 + 9y^2 = 36$