91Ó°ÊÓ

When we talk about a circular cross-section in geometry, we are referring to a slice through a three-dimensional shape that reveals a circle. In the context of the exercise, the equation \(x^2 + y^2 = 4\) is a classic representation of a circle on a two-dimensional plane, the \(xy\)-plane.
This circle has:

• A center at the origin, which is the point \((0, 0)\).
• A radius of 2. The radius is determined by the equation set to 4, so it's actually \(\sqrt{4}\).

Because this circle exists within the \(xy\)-plane, it doesn't move up or down, staying flat without any impact from the \(z\)-coordinate.
This circle is like looking exactly at a ring from above when it lies flat on the ground.

Understanding a fixed \(z\)-coordinate is crucial when describing points in space. In the exercise, the \(z = -2\) indicates that each point on the circle is anchored at the same height in three-dimensional space.
Instead of floating at various levels, or heights, all these points are fixed 2 units below the \(xy\)-plane, creating a slice that is parallel to it:

• This slice is consistent and unchanging because the \(z\)-coordinate is constant at -2.
• It essentially stretches infinitely flat along the \(xy\)-plane at the height of -2.

So, by specifying \(z = -2\), we have pinned down all points of the circle to this specific vertical level, meaning they can't rise or fall but must stay at this height.

The \(xy\)-plane is an essential reference frame in coordinates, helping us to navigate through two-dimensional and three-dimensional spaces smoothly. It acts like a stage on which geometry and points perform.
Within the context of this problem, the \(xy\)-plane is the surface on which the circular cross-section \(x^2 + y^2 = 4\) is drawn. It's defined as being horizontal, right in the middle, and typically where \(z = 0\).

• The plane is equipped with the x-axis running left to right and the y-axis running top to bottom.
• Points on this plane only require x and y coordinates, ignoring the z aspect unless specified otherwise.

When dealing with three-dimensional coordinates, the \(xy\)-plane is a slice of the entire space, offering a simple way to conceive locations and shapes. In the exercise, while the circle is taken 'down' to \(z = -2\), the circle's essential interplay with the \(xy\)-plane remains crucial to its definition.