Algebra 2 Test Questions: Identify Conics from Ax^2 + Cy^2 + ... Equations

📚 Quick Study Guide

• 🔢 Circle: The general form is $(x - h)^2 + (y - k)^2 = r^2$. Both $x$ and $y$ are squared and have equal coefficients. In the general conic equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$, $A = C$.
• 椭圆 Ellipse: The general form is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$. Both $x$ and $y$ are squared and have unequal coefficients, but $A$ and $C$ have the same sign.
• 💫 Hyperbola: The general form is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ or $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$. Both $x$ and $y$ are squared, but $A$ and $C$ have opposite signs.
• 📏 Parabola: The general form is $y = a(x - h)^2 + k$ or $x = a(y - k)^2 + h$. Only one of $x$ or $y$ is squared.
• 💡 Discriminant: You can often identify conics using the discriminant $B^2 - 4AC$ (where $B$ is the coefficient of the $xy$ term, which is 0 in these cases). If $B^2 - 4AC < 0$, it's an ellipse (or circle if $A = C$). If $B^2 - 4AC = 0$, it's a parabola. If $B^2 - 4AC > 0$, it's a hyperbola.

🧪 Practice Quiz

1. Which conic section is represented by the equation $x^2 + y^2 - 4x + 6y - 12 = 0$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola
2. Which conic section is represented by the equation $4x^2 + 9y^2 - 16x + 18y - 11 = 0$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola
3. Which conic section is represented by the equation $x^2 - y^2 + 2x - 4y - 4 = 0$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola
4. Which conic section is represented by the equation $y^2 - 8x - 6y + 1 = 0$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola
5. Which conic section is represented by the equation $25x^2 + 25y^2 + 100x - 50y - 250 = 0$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola
6. Which conic section is represented by the equation $9x^2 - 4y^2 - 18x + 8y - 31 = 0$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola
7. Which conic section is represented by the equation $y = 2x^2 - 4x + 5$?
   1. Circle
   2. Ellipse
   3. Hyperbola
   4. Parabola