To solve the problem and classify each quadratic equation in terms of the shape of the conic (circle, ellipse, hyperbola, or parabola), let's analyze each option from the image step by step:

General Conic Equation Form:

The general form of a conic section is given by:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

• If $A = C$ and $B = 0$, it's a circle.
• If $A \neq C$ and $B = 0$, it's an ellipse.
• If $A \times C < 0$, it's a hyperbola.
• If only one of $A$ or $C$ exists, it's a parabola.

Now, let's analyze each equation.

(a) $x^2 - 3y^2 - 2x + 2y + 9 = 0$

Here, $A = 1$, $B = 0$, $C = -3$.

• Since $A \times C = 1 \times (-3) = -3 < 0$, this is a hyperbola.

(b) $x^2 + 3y^2 - 2x + 2y - 9 = 0$

Here, $A = 1$, $B = 0$, $C = 3$.

• Since $A \times C = 1 \times 3 = 3 > 0$ and $A \neq C$, this is an ellipse.

(c) $2x^2 + 2y^2 + 2x - 2y - 9 = 0$

Here, $A = 2$, $B = 0$, $C = 2$.

• Since $A = C$ and $A \times C = 4 > 0$, this is a circle.

(d) $2x^2 + 2x - 2y + 9 = 0$

Here, $A = 2$, $B = 0$, $C = 0$.

• Since $C = 0$ (no $y^2$ term), this is a parabola.

Curve Types Summary:

• (a) Hyperbola
• (b) Ellipse
• (c) Circle
• (d) Parabola

Would you like me to create graphical representations of these curves or provide further details?

Relative Questions:

1. How are hyperbolas different from ellipses in terms of eccentricity?
2. Can an equation represent a degenerate conic, such as intersecting lines?
3. What real-life phenomena can be modeled using parabolas?
4. How do the coefficients of $x^2$ and $y^2$ determine the shape of the curve?
5. In what cases does a circle become an ellipse in conic sections?

Tip:

To recognize a conic section quickly, focus on the coefficients of $x^2$ and $y^2$. Equal coefficients suggest a circle, differing coefficients point to an ellipse or hyperbola, and the absence of one term suggests a parabola.