Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the equation of the circle in standard form. Then identify its center and radius. $$\frac{1}{4} x^{2}+\frac{1}{4} y^{2}=1$$

Step-by-Step Solution

Answer

The standard form of the circle's equation is \( (x - 0)^{2} + (y - 0)^{2} = 2^{2} \). Its center is at the origin (0,0) and the radius is 2.

Step 1: Get the standard form

Rewrite the given equation \( \frac{1}{4} x^{2} + \frac{1}{4} y^{2} = 1 \) as \( (x - 0)^{2} + (y - 0)^{2} = 2^{2} \). This means the h and k values (center of the circle) are both 0, and the radius is 2. Therefore, we have the standard form equation of the circle and its center.

Step 2: Identify the radius

Comparing the equation to the standard form of a circle, we gather that the square of the radius \( r^{2} = 2^{2} = 4 \). In order to get the radius we take the square root of 4, which results in 2. Therefore, the radius of the circle is 2.

Key Concepts

Standard Form Equation

Understanding the standard form of a circle's equation is crucial in identifying key aspects of the circle, such as its center and radius. The standard form of a circle's equation is written as: \[ (x - h)^2 + (y - k)^2 = r^2 \] Here,

• \((h, k)\) represents the coordinates of the circle's center.
• \(r\) stands for the radius of the circle.

The equation \( \frac{1}{4} x^2 + \frac{1}{4} y^2 = 1 \) can be rewritten in standard form as \((x - 0)^2 + (y - 0)^2 = 2^2\). This conversion helps visually understand the circle's dimensions and location. Keep in mind that rewriting the equation in standard form is a stepping stone to further analyzing the circle.

Circle Center

Identifying the center of the circle from its equation in standard form helps in understanding its position on a coordinate plane. For the standard equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] The center is located at the point \((h, k)\). In our example, the equation \((x - 0)^2 + (y - 0)^2 = 2^2\) indicates that:

• \(h = 0\)
• \(k = 0\)

Therefore, the center of the circle is at the origin of the coordinate plane, i.e., \((0, 0)\). Knowing the center aids in graphing the circle and understanding its orientation.

Circle Radius

The radius is a fundamental metric in understanding circles and is easily derived from the equation written in standard form. For the equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] The radius is given by the value of \(r\), as seen from \(r^2\). In the example \((x - 0)^2 + (y - 0)^2 = 2^2\), we identify that:

• The term \(r^2 = 2^2\)
• Taking the square root gives \(r = 2\)

Thus, the radius of the circle is 2. This length is crucial for plotting the circle accurately on a graph and understanding the circle's size.