To determine which of the given equations corresponds to a conic section formed when a plane intersects a cone parallel to the base, let’s analyze each given equation:
1. [tex]\( x^2 + y^2 = 3^2 \)[/tex]
2. [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]
3. [tex]\( x^2 = 8 y \)[/tex]
4. [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]
### Step-by-Step Analysis:
1. Equation [tex]\( x^2 + y^2 = 3^2 \)[/tex]:
- This equation is in the form of [tex]\( x^2 + y^2 = r^2 \)[/tex], which represents a circle centered at the origin with radius [tex]\(\sqrt{9} = 3\)[/tex].
- Conic section: Circle
2. Equation [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]:
- This equation is in the form of [tex]\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)[/tex], which represents an ellipse centered at the origin with semi-major axis 3 and semi-minor axis 2.
- Conic section: Ellipse
3. Equation [tex]\( x^2 = 8y \)[/tex]:
- This equation is in the form of [tex]\( x^2 = 4ay \)[/tex], which represents a parabola that opens upwards.
- Conic section: Parabola
4. Equation [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]:
- This equation is in the form of [tex]\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)[/tex], which represents a hyperbola centered at the origin.
- Conic section: Hyperbola
### Conclusion:
- A circle is specifically formed when a plane intersects a cone parallel to the base of the cone.
- Among the given equations, [tex]\( x^2 + y^2 = 3^2 \)[/tex] represents a circle.
Therefore, the correct equation corresponding to a conic section formed when a plane intersects a cone parallel to the base is:
[tex]\[ x^2 + y^2 = 3^2 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{1} \][/tex]
