Problem 60 - Algebra 2

Question

Find the center and the radius of each circle. $$ (x+2)^{2}+(y+4)^{2}=50 $$

Step-by-Step Solution

Answer

The center of the circle is \((-2, -4)\) and the radius of the circle is \(5\sqrt{2}\).

Step 1: Identify the Center Coordinates

The coordinates of the circle's center are \(-h\) and \(-k\). Therefore, by comparing the equation \( (x+2)^{2}+(y+4)^{2}=50 \) to the standard form, the coordinates of the center of the circle can be identified as \(-2\) (for \(x\)) and \(-4\) (for \(y\)). Thus, the center of the circle is at point \((-2, -4)\).

Step 2: Identify the Radius of the Circle

The radius \(r\) of the circle is obtained from the equation by taking the square root of the number on the other side of the equation, which is \(50\). Thus, the radius of the circle is \(\sqrt{50}\). Simplifying this results in \(r = 5\sqrt{2}\), where \(r\) is rounded to the nearest integer.

Key Concepts

Center of Circle

In the study of circle equations, understanding the center of a circle is crucial. The center of a circle is represented by the point \(h, k\) in the standard form equation of the circle: \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h\) and \(k\) are the coordinates of the center, while \(r\) is the radius of the circle.

To find the center of a circle from its equation, compare it with the standard form and identify the changes made to \(x\) and \(y\). These changes are reflected in the terms \((x-h)^2\) and \((y-k)^2\), where \(h\) and \(k\) are subtracted from \(x\) and \(y\) respectively. Therefore:

• For \( (x+2)^2 \), this is equivalent to \((x - (-2))^2\). Hence, \(h = -2\).
• For \( (y+4)^2 \), this equals \((y - (-4))^2\). Thus, \(k = -4\).

So, the center of the circle given by the equation \( (x+2)^2 + (y+4)^2 = 50 \) is \dash;2, dash;4). Understanding these coordinates helps in defining the location of a circle on the Cartesian plane.

Radius of Circle

The radius of a circle is a crucial aspect that defines its size. In the standard equation \((x-h)^2 + (y-k)^2 = r^2\), the expression on the right side represents \(r^2\), where \(r\) is the radius.

To determine the radius from the circle's equation, take the square root of the constant value on the right side. In the equation \( (x+2)^2 + (y+4)^2 = 50 \), the value \(50\) is equal to \(r^2\).

• Take the square root of \(50\), giving \(r = \sqrt{50}\).
• Simplify the square root to find \(r = 5\sqrt{2}\).

Therefore, the radius of the circle is \(5\sqrt{2}\), which gives you an idea of how far any point on the circle is from its center. Recognizing the radius offers valuable insights into the circle's breadth in geometry.

Standard Form of Circle Equation

The standard form of a circle equation is a powerful tool for easy understanding of its geometric properties. The standard form is written as \( (x-h)^2 + (y-k)^2 = r^2 \), where:

• \(h\) and \(k\) represent the coordinates of the center of the circle.
• \(r\) is the radius of the circle.

This format reveals all necessary information about the circle's position and size in a simple manner.

When you encounter an equation in this form, you can immediately identify key details without additional computations. For example, from \( (x+2)^2 + (y+4)^2 = 50 \), you can quickly recognize:

• The center of the circle at \((-2, -4)\).
• The radius, calculated as \(\sqrt{50}\) or \(5\sqrt{2}\).

Mastering this equation format enhances your ability to analyze and visualize circular shapes in mathematical problems.