To determine which of the given equations corresponds to a conic section formed when a plane intersects a cone at an angle to the base but not at right angles (which forms an ellipse), let's examine each equation:
1. [tex]\( x^2 + y^2 = 3^2 \)[/tex]
- This is a circle equation with radius 3, formed by setting a distance from the center point equal in all directions.
2. [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]
- This is the standard form of an ellipse equation, where the denominators [tex]\( 2^2 \)[/tex] and [tex]\( 3^2 \)[/tex] correspond to the squares of the semi-major and semi-minor axes, respectively.
3. [tex]\( x^2 = 8y \)[/tex]
- This is the standard form of a parabola equation. A parabola is a conic section formed when a plane intersects a cone parallel to the slant edge of the cone.
4. [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]
- This is the standard form of a hyperbola equation. A hyperbola is a conic section formed when a plane intersects both nappes (the upper and lower parts) of a double cone at an angle.
To identify the appropriate equation for a plane intersecting a cone at an angle to the base but not at right angles (forming an ellipse), we need to select the equation in standard ellipse form.
Thus, the equation that corresponds to this scenario is:
[tex]\[ \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \][/tex]
