The image you've uploaded shows a set of practice exercises involving conic sections. These are quadratic equations in two variables $x$ and $y$, and they need to be classified into categories such as ellipse, parabola, or hyperbola based on their general form. Here's the breakdown of the equations:

1. $-2x^2 - 5y^2 + 4x - 8y + 1 = 0$
2. $-x^2 + 4x - 3y + 2 = 0$
3. $3x^2 + 3y^2 - 8x + 4y - 2 = 0$
4. $2y^2 + 4x - 3 = 0$
5. $-6x^2 + 3y^2 - 6x + 2y - 7 = 0$

To classify them into conic sections, here is a general approach:

Step 1: Identify the coefficients

In general, the equation of a conic section is: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ Where:

• If $A \neq C$ and $B = 0$, it is a hyperbola.
• If $A = C$, it is a circle (if $A = C > 0$) or an ellipse (if $A \neq C$).
• If either $A = 0$ or $C = 0$ but not both, it is a parabola.

Step 2: Classify the given equations

1. Equation 1: $-2x^2 - 5y^2 + 4x - 8y + 1 = 0$

   • Here, $A = -2$ and $C = -5$. Since $A \neq C$ and both are negative, this is an ellipse.

2. Equation 2: $-x^2 + 4x - 3y + 2 = 0$

   • Here, $A = -1$ and there is no $y^2$ term, which makes it a parabola.

3. Equation 3: $3x^2 + 3y^2 - 8x + 4y - 2 = 0$

   • Here, $A = 3$ and $C = 3$, meaning $A = C$, so it is a circle.

4. Equation 4: $2y^2 + 4x - 3 = 0$

   • Here, there is no $x^2$ term, and only $y^2$, so this is a parabola.

5. Equation 5: $-6x^2 + 3y^2 - 6x + 2y - 7 = 0$

   • Here, $A = -6$ and $C = 3$, since $A \neq C$ and their signs are different, this is a hyperbola.

Would you like a more detailed explanation of any of these classifications, or help with solving any specific equation?

Here are five questions related to this topic:

1. How do you identify the type of conic section from the general quadratic equation?
2. What is the significance of the $Bxy$ term in the classification of conics?
3. How can you convert a general conic section equation to its standard form?
4. What are the differences in the geometrical shapes of parabolas, ellipses, and hyperbolas?
5. How do you complete the square to simplify the equation of a conic section?

Tip: Always check the signs of $A$ and $C$ to distinguish between ellipses and hyperbolas!