Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the center and radius of the circle. $$(x+1)^{2}+(y+6)^{2}=19$$

Step-by-Step Solution

Answer

The center of the circle is at (-1,-6) and the radius is \(\sqrt{19}\)

Step 1: Identify the Center of the Circle

In the standard form of a circle equation, the center (h,k) is represented by the numbers that make the expressions inside the parentheses equals zero. Our equation is \((x+1)^{2}+(y+6)^{2}=19\), thus by setting \(x+1 = 0\) and \(y+6 = 0\), it gives \(x = -1\) and \(y = -6\). Therefore, the center (h,k) of the circle is (-1,-6)

Step 2: Identify the Radius

The radius is given by the square root of the number on the right side of the equation, it's \(\sqrt{r^{2}}\). Thus, the radius \(r\) of the circle is \(\sqrt{19}\)

Key Concepts

Center of a Circle

The center of a circle is a crucial concept when studying circle equations. In the standard form of a circle's equation, \((x-h)^2 + (y-k)^2 = r^2\), the center is represented by the coordinates \((h, k)\). To find the center:

• Identify the numbers inside the parentheses next to \(x\) and \(y\).
• Set those expressions equal to zero: \(x - h = 0\) and \(y - k = 0\).

For example, in the equation \((x+1)^2 + (y+6)^2 = 19\), setting \(x+1 = 0\) and \(y+6 = 0\) gives \(x = -1\) and \(y = -6\), so the center is \((-1, -6)\).
Finding the center is like finding the middle point of the circle where all points are equidistant.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. In the standard form equation:

• The term \(r^2\) on the right side is the square of the radius.
• Simply take the square root of this number to find the radius \(r\).

For example, in the equation \((x+1)^2 + (y+6)^2 = 19\), the number \(19\) represents \(r^2\). Taking the square root, the radius \(r\) is \(\sqrt{19}\).
The radius is vital for understanding the size of the circle and is used in many applications, such as calculating circumference and area.

Standard Form

The standard form of a circle's equation simplifies the process of identifying its center and radius. This form is expressed as: \[(x-h)^2 + (y-k)^2 = r^2\]Here’s how to use it:

• \(h\) and \(k\) are the coordinates of the center.
• \(r\) is the radius, found by taking the square root of \(r^2\).

For instance, the equation \((x+1)^2 + (y+6)^2 = 19\) fits this format, allowing us to easily extract the center \((-1, -6)\) and the radius \(\sqrt{19}\).
Using the standard form makes these calculations straightforward, providing a clear framework for analyzing a circle's properties.