Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-34: Sketch the graph of the circle or semicircle. $$ 4 x^{2}+4 y^{2}=25 $$

Step-by-Step Solution

Answer

The graph of \(4x^{2} + 4y^{2} = 25\) is a circle with center (0,0) and radius \(\frac{5}{2}\).

Step 1: Identify Circle Equation

The given equation of the geometrical shape is \(4x^{2} + 4y^{2} = 25\). This equation represents a circle, but it is not in the standard form of a circle, which is \((x-h)^2 + (y-k)^2 = r^2\). To rewrite it, observe that both \(x^2\) and \(y^2\) are multiplied by 4.

Step 2: Simplify to Standard Form

Divide every term in the equation by 4 to simplify it: \(x^{2} + y^{2} = \frac{25}{4}\). This is now in the standard form of a circle \((x-h)^2 + (y-k)^2 = r^2\), where \(h = 0\), \(k = 0\), and \(r^2 = \frac{25}{4}\). Thus, the center of the circle is at the origin (0,0) and the radius \(r = \sqrt{\frac{25}{4}} = \frac{5}{2}\).

Step 3: Plot the Circle

On a coordinate plane, place a point at the origin to represent the center of the circle. Using the radius \(\frac{5}{2}\), measure out from the origin in all directions to the points (\(\frac{5}{2},0\)), (-\(\frac{5}{2},0\)), (0,\(\frac{5}{2}\)), and (0,-\(\frac{5}{2}\)). These are the endpoints of the circle's diameter lines on the x and y axes.

Step 4: Sketch the Circle

Draw a smooth round curve connecting these four points, making sure the curve is equidistant from the center at all points. This completes the sketch of the circle defined by the simplified equation.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is expressed as \((x-h)^2 + (y-k)^2 = r^2\). This format is particularly useful because it provides immediate information about the circle's position and size:

• \((h, k)\) represents the center of the circle.
• \(r^2\) is the radius squared.

By transforming an equation to this form, you make it easier to understand and graph the circle. In many exercises, including the one here, you begin by rewriting a given equation to match this standard form. Doing so first involves rearranging and simplifying the given terms. For example, when we have the equation \(4x^2 + 4y^2 = 25\), we recognize that both \(x^2\) and \(y^2\) are multiplied by 4. To transform to the standard form, divide every term by 4, yielding \(x^2 + y^2 = \frac{25}{4}\). This then clearly indicates the circle's equation in the standard format.

Center and Radius of a Circle

Once we have a circle's equation in the standard form \((x-h)^2 + (y-k)^2 = r^2\), identifying the center and radius is a straightforward task. The

• center is the point \((h, k)\), where \(h\) and \(k\) are constants from the equation.
• radius is \(r\), which is the square root of \(r^2\).

In the transformed equation \(x^2 + y^2 = \frac{25}{4}\), we see there are no extra \(h\) or \(k\) terms, implying \(h = 0\) and \(k = 0\). Thus, the center is at the origin (0,0), which simplifies graphing considerably.
The process of obtaining the radius involves solving \(r = \sqrt{r^2}\). With the solution's equation, this means calculating \(r = \sqrt{\frac{25}{4}}\), resulting in a radius \(r = \frac{5}{2}\). Knowing the center and radius, we have all we need to graph the circle.

Graphing Circles

When it comes to graphing circles, positioning the center and using the radius efficiently are vital steps. With the exercise's equation reformed to \(x^2 + y^2 = \frac{25}{4}\), the center of the circle is already identified at the origin (0,0) on the coordinate plane.
To graph:

• Start at the center point and mark it clearly on your plane.
• Using the radius \(\frac{5}{2}\), measure this distance outwards from the origin in four cardinal directions: right and left along the x-axis and up and down along the y-axis.
• This will create four key points at \((\frac{5}{2}, 0), (-\frac{5}{2}, 0), (0, \frac{5}{2}), (0, -\frac{5}{2})\).
• Finally, draw a smooth, round curve connecting these points, ensuring it's equidistant from the center to maintain the circle's shape.

These steps lead to a neat and accurate representation of the circle on your graph, completing the process visually and reinforcing comprehension of standard circle equations.