Problem 60 - Calculus for Biology and Medicine

Question

Find the center and the radius of the circle given by the equation $$ (x+1)^{2}+(y-3)^{2}=9 $$

Step-by-Step Solution

Answer

Center: (-1, 3); Radius: 3.

Step 1: Identify the standard form of a circle equation

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

Step 2: Compare with the given equation

Look at the given equation \((x+1)^2 + (y-3)^2 = 9\) and compare it with the standard form. You can identify that \((h, k) = (-1, 3)\).

Step 3: Find the center of the circle

From the standard form, the center of the circle \((h, k)\) can be identified. Using our comparison, the center is \((-1, 3)\).

Step 4: Identify the radius squared

The given circle equation is \((x+1)^2 + (y-3)^2 = 9\). From this, \(r^2 = 9\).

Step 5: Calculate the radius

To find the radius, take the square root of \(9\). Thus, \(r = \sqrt{9} = 3\).

Key Concepts

Center of a Circle

The center of a circle is a key concept in geometry that helps us understand the position of the circle on a coordinate plane. When given the equation of a circle in its standard form, \((x-h)^2 + (y-k)^2 = r^2\), the center is located at the point \((h, k)\). This simply means that the circle is centered on the point \(h\) along the x-axis and \(k\) along the y-axis.
It's similar to finding a "home base" for the circle. Let's break it down with the equation in our example, \((x+1)^2 + (y-3)^2 = 9\), and compare it to the standard form:

• Recognize that the equation components, \((x+1)\) and \((y-3)\), can be transformed to match \((x-h)\) and \((y-k)\), by rewriting them as \((x - (-1))\) and \((y - 3)\).
• This lets us see that the center of our circle is at \((-1, 3)\).

You just need to "flip the signs" inside the brackets to uncover the coordinates of the circle's center.

Radius of a Circle

The radius of a circle is another crucial element, letting us know how big or small the circle is. Mathematically speaking, the radius extends from the center of the circle to any point along its edge.
If the circle's equation is given in the form \((x-h)^2 + (y-k)^2 = r^2\), then \(r\) represents the radius.
In our example, the challenge is to determine \(r\) from the equation \((x+1)^2 + (y-3)^2 = 9\):

• The value on the right side of the equation, 9, corresponds to \(r^2\), which is the radius squared.
• To get the actual radius, take the square root of 9, leading us to \(r = \sqrt{9} = 3\).

The radius tells us not just the size of the circle but also provides insight into how far any point on the circle is from its center.

Standard Form of a Circle's Equation

The standard form of a circle's equation is a convenient way to easily identify crucial properties such as the center and the radius of the circle.
This form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
It organizes information in a neat way, giving you quick access to the circle's core details:

• The parts \((x-h)\) and \((y-k)\) point out the center coordinates by shifting the signs. \(h\) and \(k\) are found by applying opposite signs to the values enclosed.
• The right side of the equation, \(r^2\), helps us find the radius by taking its square root.

Using this form, equations of circles become straightforward to work with, without turning the math into a puzzle. Remember, once the equation is set in this standard form, identifying the circle's center and radius becomes a breeze.