Problem 93 - Contemporary Precalculus

Question

For each nonzero real number \(k\), the graph of \((x-k)^{2}+y^{2}=k^{2}\) is a circle. Describe all possible such circles.

Step-by-Step Solution

Answer

Question: Describe all possible circles with the equation \((x-k)^2 + y^2 = k^2\), where k is a nonzero real number. Answer: All possible circles described by the given equation have centers on the x-axis (excluding the origin) and their radii equal their x-coordinate. No circles have centers in quadrants 2 and 4, and circles with centers in quadrants 1 and 3 will not intersect the y-axis.

Step 1: Identify the circle's equation

The given equation is \((x-k)^2 + y^2 = k^2\). This is the equation of a circle in the standard form, which can be written as \((x-a)^2 + (y-b)^2 = r^2\), where (a,b) is the center of the circle and r is its radius.

Step 2: Identify the center and radius of the circle

From the given equation, we can deduce that the center of the circle is given by (a,b) = (k,0) and its radius is r = k. Notice that k is both the x-coordinate of the center and the radius of the circle.

Step 3: Describe all possible circles

Since k can be any nonzero real number, the x-coordinate of the circle's center can be any nonzero number. The y-coordinate will always be 0 because the equation is in terms of y^2 only. The radius of the circle is equal to the x-coordinate of its center. Therefore, all possible circles described by the given equation have centers on the x-axis (excluding the origin) and their radii equal their x-coordinate. This means there are no circles with centers in quadrants 2 and 4, and circles with centers in quadrants 1 and 3 will not intersect the y-axis.

Key Concepts

Circle Standard Form Equation

Understanding the standard form equation of a circle is crucial for analyzing and graphing circles in precalculus. It's the foundation from which we can discern the properties of a circle such as its center, radius, and position on the coordinate plane. The standard form of a circle's equation is (x - a)^2 + (y - b)^2 = r^2, where (a, b) represents the center of the circle and r stands for its radius.

When we examine an equation like (x - k)^2 + y^2 = k^2, it aligns with the standard form, suggesting a circle with a center and radius based on the value of k. To help with clarity and understanding, consider reworking practice problems to reflect different values of k and sketching the resultant circles. This visualization can strengthen comprehension of how changes in the equation reflect changes in the circle's graph.

Circle Center and Radius

The center and radius of a circle are among the most pivotal characteristics to grasp. From the equation (x - k)^2 + y^2 = k^2, by drawing parallels to the standard form, we can ascertain that the center point is (k, 0), lying on the x-axis. Moreover, the radius equates to the value of k. One may be mistaken to think that the circle can be centered at the origin or have a zero radius, but it's imperative to remember that k represents nonzero real numbers. This precludes the possibility of the origin being the center.

The center's fixed y-coordinate of zero indicates that all circles will be above or below the origin, yet they will never cross the y-axis. As we consider different values for k, picturing where each corresponding circle would reside on a graph sheet can aid students significantly in visualizing and understanding these concepts in a more concrete manner.

Nonzero Real Numbers

The term 'nonzero real numbers' plays an integral role in the variety of circles one might describe with the equation (x - k)^2 + y^2 = k^2. Nonzero means the numbers are not zero; they can be positive or negative but must exist on the number line. This restriction ensures that the center (k, 0) is never at the origin and that the radius k is never zero, thus always producing a tangible circle on the graph.

By excluding zero, we limit the potential centers to lie exclusively in quadrants 1 and 3 of the coordinate plane. The radius similarly correlates to the absolute value of k, furnishing us with circles that vary in size but persist in having congruence between their radius and the x-coordinate of the center. A practical tip for students struggling with this concept is to generate a list of possible values for k and plot the resulting circles to demonstrate the diversity and limitations imposed by being a nonzero real number.