Question

a plane intersects a cone parallel to the base of the cone. which graph shows the conic section created by this intersection? 3 of 4 questions

Explanation:

When a plane intersects a cone parallel to the base of the cone, the conic - section formed is a circle.

1. Hyperbola (first option):

• A hyperbola is formed when a plane intersects both nappes of a double - napped cone. The equation of a hyperbola is of the form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\) (for a hyperbola centered at the origin with a horizontal transverse axis) or \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) (for a hyperbola centered at the origin with a vertical transverse axis). The graph of a hyperbola has two separate branches.

1. Ellipse (second option):

• An ellipse is formed when a plane intersects a single nappe of a cone at an angle (other than parallel to the base or parallel to the side of the cone). The standard form of an ellipse centered at the origin is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (\(a

eq b\)).

1. Parabola (third option):

• A parabola is formed when a plane intersects a single nappe of a cone parallel to the side of the cone. The standard form of a parabola is \(y=ax^{2}+bx + c\) (for a parabola that opens up or down) or \(x = ay^{2}+by + c\) (for a parabola that opens left or right).

1. Circle (fourth option):

• A circle is a special case of an ellipse where \(a = b\). When a plane is parallel to the base of the cone, the intersection satisfies the equation of a circle \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Answer:

The fourth option (the circle)