Problem 77 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$5 x^{2}+5 y^{2}=25$$

Step-by-Step Solution

Answer

The graph is a circle with center (0,0) and radius \(\sqrt{5}\).

Step 1: Convert Equation to Standard Form

Start with the given equation: \(5x^2 + 5y^2 = 25\). Divide each term by 5 to simplify the equation to \(x^2 + y^2 = 5\).

Step 2: Identify the Graph Type

The simplified equation \(x^2 + y^2 = 5\) represents a circle equation in the standard form \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step 3: Determine the Center of the Circle

Compare the equation \(x^2 + y^2 = 5\) with the circle's standard form. Since there are no terms implying shifts of the center (like \(x-h\) or \(y-k\)), the center of the circle is at the origin \((0, 0)\).

Step 4: Find the Radius of the Circle

The right side of the circle's standard form, \(r^2\), equals 5 in the equation \(x^2 + y^2 = 5\). Therefore, the radius \(r\) is \(\sqrt{5}\).

Step 5: Sketch the Graph

Draw a circle centered at \((0, 0)\). Since \(r = \sqrt{5}\), which is approximately \(2.24\), make the circle pass through points approximately 2.24 units away from the origin, like \((\sqrt{5}, 0)\) and \((0, \sqrt{5})\).

Key Concepts

Circle Equation

Understanding the equation of a circle is foundational in the study of conic sections. When you are given an equation, such as \(5x^2 + 5y^2 = 25\), this typically represents a circle when written in the standard form. To make the equation more recognizable, we simplify it.
This involves dividing each term by the same factor, here it is 5, leading to the equation \(x^2 + y^2 = 5\). This is the standard form of a circle's equation, \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) are the coordinates of the circle's center and \(r\) is the radius.
In our case, the absence of coefficients next to the \(x\) and \(y\) terms, other than 1s, indicates the circle is centered at the origin, and the equation on the right, 5, represents \(r^2\). This provides a clear formula to identify the basic properties of any circle, namely its center and radius.
To summarize:

• Divide to simplify the equation
• Recognize the standard form of \( (x-h)^2 + (y-k)^2 = r^2 \)
• Identify the absence of adjustments from \((x-h)\) or \((y-k)\) as indicating the center \( (0,0)\)

Graphing Circles

Graphing a circle effectively begins with mastering its equation. For the equation \(x^2 + y^2 = 5\), the task is to represent this geometrically on a coordinate plane.
First, identify the center as the point where the circle will be anchored. In this instance, it's \((0, 0)\), indicating the circle is based at the origin with symmetry in all directions. The radius is determined by taking the square root of the number on the right side of the equation. Here, \(r = \sqrt{5}\), which is approximately 2.24.
When sketching, consider these steps:

• Mark the center at \((0, 0)\)
• From the center, measure the radius outwards in each of the quadrant directions
• Ensure the circle passes through points like \((\sqrt{5}, 0)\) and \((0, \sqrt{5})\)

Remember to achieve a smooth, round shape, as it's not a polygon but a continuous curve with every point equidistant from the center.

Center and Radius

The concepts of the center and radius are crucial for analyzing and graphing circles. For the equation \(x^2 + y^2 = 5\), both these elements become straightforward to define.

• Center: In the standard form \((x-h)^2 + (y-k)^2 = r^2\), the \(h\) and \(k\) denote the coordinates of the center. In our given equation, with \(h = 0\) and \(k = 0\), the center lies at the origin \((0, 0)\).
• Radius: The term on the right of the equation, \(r^2 = 5\), allows for the calculation of the radius. The radius \(r\) is found by taking the square root of 5, giving us approximately 2.24 units. This measurement from the center stretches to all points along the circle's circumference.

Understanding these helps not only in graphing but in interpreting the geometric significance of a circle's equation, providing insights into its size and position on the coordinate plane.