Question

Sketch the \(x y\) -trace of the sphere. $$ (x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25 $$

Short answer

The x-y trace of the sphere is a circle with a center at (1,3) and a radius of \(\sqrt{21}\).

Step-by-step solution

Set z to zero

For getting the x-y trace, we have to consider that z=0. So, replace z in the given equation with zero. The sphere equation then becomes \((x-1)^{2}+(y-3)^{2}+ (2)^{2}=25\).

Simplify the equation

Solve this equation to get the x-y trace which is a circle. The equation becomes \((x-1)^{2}+(y-3)^{2}= 21\).

Identify the circle properties

From the equation \((x-1)^{2}+(y-3)^{2}=21\), we can say that the circle's center is at point (1,3) and its radius is \(\sqrt{21}\).

Sketch the circle

Using the properties, sketch the circle on the x-y plane. The circle is centered at (1,3) and has a radius of \(\sqrt{21}\).

Key concepts

Sphere Equation

Understanding the equation of a sphere is crucial before we can delve into sketching its traces. A sphere's equation in algebra is typically given in the standard form \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \), where \( (h, k, l) \) represents the coordinates of the sphere’s center and \( r \) is the radius.

Every term in the equation has its significance. The variables \( x, y, \) and \( z \) correspond to the coordinates of any point on the surface of the sphere, while \( h, k, \) and \( l \) give us the precise location of the center.

To sketch a sphere from its equation, one must recognize that the power of 2 on each term signifies that every point on the sphere is equidistant from the center—this distance being the radius of the sphere.

X-Y Trace

The x-y trace of a sphere is a useful concept for visualizing the cross-section of the sphere when sliced by the plane \( z=0 \). It is essentially the intersection of the sphere with the x-y plane.

To find this trace, we set \( z = 0 \) in the equation of the sphere, simplifying the three-dimensional situation into a more manageable two-dimensional problem. What remains is the equation of a circle, which corresponds to the outline of the sphere's shadow on the x-y plane.

Thus, for the given sphere equation \( (x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25 \), by substituting \( z=0 \) we're left with \( (x-1)^{2}+(y-3)^{2}=21 \), which represents the x-y trace, a circle with properties that can be extracted for sketching.

Circle Properties

When we talk about the properties of a circle, we are referring to the various characteristics that define a specific circle, such as its radius, diameter, and center. The generic equation for a circle in the plane is \( (x-a)^2 + (y-b)^2 = r^2 \) where \( (a, b) \) is the center and \( r \) is the radius.

For instance, from the circle equation \( (x-1)^2 + (y-3)^2 = 21 \) obtained from the x-y trace of the given sphere, we can deduce that the center of this circle is \( (1,3) \) and the radius is \( \sqrt{21} \).

The radius is a straight line from the center to any point on the circle's edge, and its length is paramount as it determines all other metrics of the circle such as circumference and area. The center and radius together allow us to accurately sketch and identify the circle's position in the plane.

Sketching Circles in Algebra

Sketching circles in algebra is about transforming an algebraic equation into a visual graph. Once a circle's equation is simplified, and its properties like the center and radius identified, sketching it becomes an application of these properties.

Starting with the center \( (1,3) \) for our circle from the x-y trace of the sphere, we plot this point on the x-y plane. Next, using the radius \( \sqrt{21} \), we measure outwards in all directions from the center to mark the edge of the circle. Then, we draw a smooth curve connecting these points to form the circumference of the circle.

It's essential to ensure the curve is uniform and round as any deviation might represent a different geometric figure. For enhanced precision, a compass or a circle drawing tool is preferable. This process, when done carefully, will yield an accurate representation of the circle on the graph, showcasing the relationship between algebraic formulas and their geometric counterparts.