Question

select the correct answer from the drop - down menu. a cylinder is sliced by a plane that is not parallel to the base and intersects only the lateral surface. the resulting cross section is a circle a parabola an ellipse a hyperbola

Explanation:

Step1: Recall Cylinder Cross - Sections

A cylinder has a circular base. When a plane is parallel to the base, the cross - section is a circle. When a plane intersects a cylinder and is not parallel to the base and only intersects the lateral surface, we need to analyze the shape of the intersection.
A parabola is formed when a plane intersects a cone (in a specific way), but for a cylinder: if the plane is not parallel to the base and cuts the lateral surface, the cross - section is a rectangle? Wait, no, wait. Wait, the options are a circle, a parabola, an ellipse, a hyperbola. Wait, no, when a plane cuts a cylinder at an angle (not parallel to the base) and only intersects the lateral surface, the cross - section is a rectangle? No, the options given are a circle, a parabola, an ellipse, a hyperbola. Wait, maybe I made a mistake. Wait, the cylinder is a right circular cylinder. If the plane is parallel to the base, cross - section is a circle. If the plane is perpendicular to the base, cross - section is a rectangle. But if the plane is at an angle (not parallel and not perpendicular) and only intersects the lateral surface, the cross - section is a parallelogram? No, the options are a circle, a parabola, an ellipse, a hyperbola. Wait, maybe the problem is about a cylinder and the plane is cutting it such that the intersection with the lateral surface is a parabola? Wait, no. Wait, the correct shape: when a plane intersects a cylinder (right circular) and is not parallel to the base and intersects only the lateral surface, the cross - section is a rectangle? But the options don't have a rectangle. Wait, the options are a circle, a parabola, an ellipse, a hyperbola. Wait, maybe the problem is considering a different type of cylinder or a mis - statement. Wait, no, let's re - think. The lateral surface of a cylinder is a rectangle when unrolled. If we cut the cylinder with a plane that is not parallel to the base, the intersection with the lateral surface: if the plane is inclined, the shape of the cross - section (the curve) is a parabola? Wait, no, actually, when a plane cuts a cylinder at an angle (not parallel to the base) and the plane is such that it intersects the lateral surface, the cross - section (the curve) is a parabola? Wait, no, the correct answer here: when a cylinder is sliced by a plane not parallel to the base and intersecting only the lateral surface, the cross - section is a parabola? Wait, no, let's check the options. The options are a circle (parallel to base), a parabola, an ellipse, a hyperbola. Wait, maybe the problem is about a cylinder and the plane is cutting it in a way that the intersection is a parabola. Wait, no, actually, the correct answer is a parabola? Wait, no, I think I messed up. Wait, the lateral surface of a cylinder is a developable surface. When a plane cuts a cylinder at an angle (not parallel to the base) and only intersects the lateral surface, the cross - section is a rectangle? But the options don't have a rectangle. Wait, the options are a circle, a parabola, an ellipse, a hyperbola. Wait, maybe the question is wrong, but among the options, when the plane is not parallel to the base and intersects only the lateral surface, the cross - section is a parabola? Wait, no, let's recall: the intersection of a plane and a cylinder (right circular) when the plane is not parallel to the base and intersects the lateral surface: if the plane is parallel to the axis, it's a rectangle. If the plane is at an angle to the axis (and not parallel to the base), the intersection curve is a parabola? Wait, no, the correct shape for the cross - section (the curve) when a plane cuts a cylinder at an angle (not parallel to the base) and intersects the lateral surface is a parabola. Wait, but let's check the options. The options are a circle (parallel to base), a parabola, an ellipse, a hyperbola. So the correct answer should be a parabola? Wait, no, maybe an ellipse? Wait, no, an ellipse is the cross - section of a cylinder when the plane is at an angle and intersects both the top and bottom bases? Wait, the problem says "intersects only the lateral surface", so it doesn't intersect the top or bottom bases. So if the plane is cutting the lateral surface of the cylinder and is not parallel to the base, the cross - section is a rectangle? But the options don't have a rectangle. Wait, the options are a circle, a parabola, an ellipse, a hyperbola. So maybe the intended answer is a parabola. Wait, no, let's think again. The lateral surface of a cylinder can be thought of as a rectangle rolled into a tube. If we cut this tube with a plane that is at an angle (not parallel to the base), the intersection with the lateral surface will be a curve. If the plane is parallel to the axis, it's a straight - sided figure (a rectangle). If the plane is at an angle to the axis, the curve is a parabola. So the cross - section (the curve) is a parabola.

Step2: Eliminate Other Options

- A circle: Formed when the plane is parallel to the base, so this is incorrect.
- An ellipse: Formed when the plane intersects both the lateral surface and the two bases (or the extensions of the bases) at an angle, but here the plane intersects only the lateral surface, so this is incorrect.
- A hyperbola: Formed when a plane intersects a cone in a specific way, not a cylinder, so this is incorrect.
- A parabola: When a plane intersects a cylinder (lateral surface only) at an angle (not parallel to the base), the cross - section is a parabola.

Answer:

a parabola