Problem 21 - Intermediate Algebra

Question

Find the center and radius of each circle and graph it. $$ x^{2}+y^{2}=10 $$

Step-by-Step Solution

Answer

Center: (0, 0); Radius: \( \sqrt{10} \)

Step 1: Identify the Circle Equation

The given equation is in the form of a standard circle equation, which is \( x^2 + y^2 = r^2 \). The center of the circle is \((h, k)\) where \(h = 0\) and \(k = 0\). The equation here is \( x^{2} + y^{2} = 10 \).

Step 2: Determine the Center

For the equation \( x^2 + y^2 = r^2 \), the coefficients of both \( x \) and \( y \) are 0, meaning the circle is centered at the origin. Thus, the center of this circle is at (0, 0).

Step 3: Compute the Radius

In the standard circle equation \( x^2 + y^2 = r^2 \), 10 is the value of \( r^2 \). Taking the square root, \( \), we find that \( r = \sqrt{10} \). Therefore, the radius of the circle is \( \sqrt{10} \).

Step 4: Graph the Circle

Plot the center of the circle at the origin (0, 0) on the Cartesian plane. Use a compass or indicate measurements to draw a circle with a radius of approximately 3.16 (since \( \sqrt{10} \approx 3.16 \)). This completes the graphical representation.

Key Concepts

Center of a Circle

Understanding the center of a circle is foundational when interpreting the equation of a circle. In the standard form equation of a circle,

• the center is denoted as \((h, k)\).
• each term being part of the shifted variables as \((x - h)^2 + (y - k)^2 = r^2\).

In this particular exercise, we see the equation is \(x^2 + y^2 = 10\). Here, there are no \((h, k)\) terms associated with \(x\) and \(y\), implying:

• the center of this circle is at the origin, \((0, 0)\).

Knowing the center helps you understand where the circle is placed on the Cartesian coordinate system. The center acts as a pivotal point from which the rest of the circle extends.

Radius Calculation

Calculating the radius from the circle's equation is a straightforward process. In the standard equation format \((x - h)^2 + (y - k)^2 = r^2\), the term \(r^2\) at the end represents the square of the radius. In our exercise:

• the equation \(x^2 + y^2 = 10\) equates to \(r^2 = 10\).

To find the radius, you extract the square root from \(r^2\):

• \(r = \sqrt{10}\).
• The numerical value approximately is 3.16.

This value signifies the distance from the circle's center to any point on its boundary. Realizing this measurement is crucial for accurate graphical representation and applications in geometry, like finding the circumference or disc area.

Graphing a Circle

Graphing a circle involves plotting all points equidistant from a central point on a graph. Knowing the center and the radius:

• Center: the point, here \((0, 0)\), acts as a pivot from where you measure the radius.
• Radius: here \(\sqrt{10} \approx 3.16\), tells you how far each point on the circle's perimeter is from the center.

To graph:- Begin at the origin (0, 0). This point is the center of your circle.- Use a compass with a measured radius of 3.16 units, or mathematically calculate points on the grid at this distance away.- Plot several points all around the center, ensuring they maintain this distance.- Draw a smooth curve through these points.
Graphing ensures you visualize the abstract equations into tangible shapes, helping in understanding spatial relations in geometry.