Problem 19 - Algebra for College Students

Question

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ 3(x-10)^{2}+3(y+5)^{2}=9 \quad(10,-5) ; r=\sqrt{3} $$

Step-by-Step Solution

Answer

Center: (10, -5); Radius: \(\sqrt{3}\).

Step 1: Identify the Standard Form of the Circle Equation

The equation given is \(3(x-10)^{2}+3(y+5)^{2}=9\). A circle's equation in standard form is \((x-h)^2 + (y-k)^2 = r^2\). To convert the given equation, divide every term by 3 to isolate the squared terms.

Step 2: Simplify the Equation

Divide the entire equation by 3: \( (x-10)^2 + (y+5)^2 = \frac{9}{3} \). Simplify this to obtain: \( (x-10)^2 + (y+5)^2 = 3 \).

Step 3: Identify the Center of the Circle

In the standard form \((x-h)^2 + (y-k)^2 = r^2\), the center of the circle is \((h, k)\). From the simplified equation \((x-10)^2 + (y+5)^2 = 3\), we identify \(h = 10\) and \(k = -5\). Thus, the center is \((10, -5)\).

Step 4: Determine the Radius of the Circle

The term \(r^2\) in the standard form is the constant on the right side of the equation. From \((x-10)^2 + (y+5)^2 = 3\), identify \(r^2 = 3\). To find \(r\), take the square root of 3, hence \(r = \sqrt{3}\).

Key Concepts

Center of the Circle

To figure out the center of a circle from its equation, we use the format

• The standard form of a circle's equation looks like this: \((x-h)^2 + (y-k)^2 = r^2\)
• Here, \(h\) and \(k\) are numbers representing the coordinates of the point at the center, also known as \((h, k)\).

In simpler words,

• The values of \(h\) and \(k\) are what place the circle on the coordinate grid, centering it at \((h, k)\).

For example, in the equation \((x-10)^2 + (y+5)^2 = 3\), the center is found by identifying the constants subtracted from \(x\) and added to \(y\).

• This gives you the center of this circle: \((10, -5)\).

The key takeaway is to focus on the numbers around \(x\) and \(y\) in the equation format \((x-h)^2 + (y-k)^2 = r^2\).

Those numbers lead you directly to the center of the circle.

Radius of a Circle

The radius is the distance from the center of the circle to any point on its outer edge.

• In the standard circle equation format \((x-h)^2 + (y-k)^2 = r^2\), \(r^2\) is the part of the equation on the right that tells us about the size of the circle.
• To find the actual radius (the distance value \(r\)), we need to take the square root of that number.

In our simplified example,

• We have \((x-10)^2 + (y+5)^2 = 3\), which means \(r^2 = 3\).
• To find \(r\), calculate \(r = \sqrt{3}\).

The radius is a crucial part of understanding a circle's size in any graph or geometric problem.

All you really need to do is look for the number on the right side of the equation and take its square root. That gives you the circle's radius.

Standard Form of a Circle Equation

Understanding circle equations starts here. The standard form of a circle's equation is your go-to method for identifying how a circle will look on a graph. It follows the structure:

• \((x-h)^2 + (y-k)^2 = r^2\)
• This setup splits into its components like the center \((h, k)\) and radius \(r\).

But sometimes, circle equations might not look like this right away. You may need to tinker with or simplify them a bit to reveal this form.

An example transformation can be seen in the given problem. Initially, it's \(3(x-10)^2 + 3(y+5)^2 = 9\), not quite matching \((x-h)^2 + (y-k)^2 = r^2\).

• Dividing everything by 3 gives \((x-10)^2 + (y+5)^2 = 3\).

You're left with a clean standard circle equation, easy to interpret on a coordinate plane. With this form, you can swiftly identify the center and the size of your circle, making it a crucial skill for any math enthusiast.