Problem 31 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at \(\left(\frac{2}{3},-\frac{7}{8}\right) ;\) radius \(\sqrt{2}\)

Step-by-Step Solution

Answer

The equation of the circle is \((x - \frac{2}{3})^2 + (y + \frac{7}{8})^2 = 2\).

Step 1: Understand the Standard Form Equation of a Circle

The standard form of the equation of a circle is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius of the circle.

Step 2: Substitute the Given Center into the Equation

Substitute the center \((h, k) = \left(\frac{2}{3}, -\frac{7}{8}\right)\) into the equation. This gives the equation as \((x - \frac{2}{3})^2 + (y + \frac{7}{8})^2 = r^2\).

Step 3: Substitute the Given Radius Square into the Equation

The given radius is \(\sqrt{2}\). The equation requires the radius squared, \(r^2\), to be substituted. Since the radius is \(\sqrt{2}\), \(r^2\) becomes \(2\).

Step 4: Write the Final Equation of the Circle

Substitute the value of \(r^2\) into the equation obtained in Step 2 to get \((x - \frac{2}{3})^2 + (y + \frac{7}{8})^2 = 2\). This is the equation of the circle in standard form with the given properties.

Key Concepts

Standard Form of a Circle

In mathematics, the standard form equation of a circle allows us to easily understand the circle's essential properties through a simple expression. The standard form of the equation of a circle is expressed as

• \((x - h)^2 + (y - k)^2 = r^2\)

Here,

• \((h, k)\) represents the center of the circle.
• \(r\) denotes the radius of the circle.

This formula is incredibly useful for identifying a circle's location and size quickly. By looking at this equation, one can immediately determine both the circle's center and its radius. This makes the standard form an essential tool in both geometry and algebra.
To solve problems involving circles, you'll often start by ensuring your information fits this formula, allowing you to plug in known values and solve for unknowns.

Center of a Circle

The center of a circle is a fundamental feature defined in the standard form equation. The center, represented as \((h, k)\), determines where the circle is positioned on the coordinate plane.
In our exercise, the center is given as \(\left(\frac{2}{3}, -\frac{7}{8}\right)\). This means that:

• \(h = \frac{2}{3}\)
• \(k = -\frac{7}{8}\)

These values are substituted directly into the standard form equation to anchor the circle precisely at the specified coordinates.
When plotting the center, note which quadrant it lies in by considering the signs of \(h\) and \(k\). The center is crucial because it serves as a reference point for measuring the radius and drawing the circle.

Radius of a Circle

The radius is another critical component in describing a circle's size. In the standard form equation, the radius \(r\) is squared to provide the value for \(r^2\). It's essential to substitute the squared radius in the equation properly.
In the given problem, the radius \(r\) is \(\sqrt{2}\). When you square \(\sqrt{2}\), you obtain 2. Therefore, \(r^2 = 2\).
Understanding how to handle the radius is key to correctly writing and interpreting the circle's equation. Remember, the radius represents the distance from the circle's center to any point on its boundary, and it's always a positive number. This distance helps dictate the circle's size on the graph and plays a significant role in comprehending its overall geometry.