Problem 40 - Precalculus

Question

Specify the center and radius of each circle. Also, determine whether the given point lies on the circle. $$(x+4)^{2}+(y+2)^{2}=20 ;(0,1)$$

Step-by-Step Solution

Answer

The circle's center is (-4, -2) and radius is \(2\sqrt{5}\). The point (0, 1) does not lie on the circle.

Step 1: Identify the Standard Equation of a Circle

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. We compare the given equation \((x + 4)^2 + (y + 2)^2 = 20\) with this standard form.

Step 2: Determine the Center of the Circle

Compare the given equation \((x + 4)^2 + (y + 2)^2 = 20\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\). Here, \(x+4\) mirrors \(x-h\), so \(h = -4\). Similarly, \(y+2\) mirrors \(y-k\), so \(k=-2\). Thus, the center of the circle is \((-4, -2)\).

Step 3: Calculate the Radius of the Circle

From the equation \((x+4)^2 + (y+2)^2 = 20\), recognize that \(r^2 = 20\). The radius \(r\) is found by taking the square root, giving \(r = \sqrt{20} = 2\sqrt{5}\).

Step 4: Verify if the Given Point Lies on the Circle

Check if the point \((0, 1)\) satisfies the circle's equation. Substitute \(x = 0\) and \(y = 1\) into the equation: \((0 + 4)^2 + (1 + 2)^2 = 16 + 9 = 25\). Since 25 is not equal to 20, the point \((0, 1)\) does not lie on the circle.

Key Concepts

Center of a Circle

Understanding the center of a circle is crucial when dealing with the circle's geometry. The center is the fixed point around which all points on the circle are equidistant. In the standard form equation of a circle, which is \((x-h)^2 + (y-k)^2 = r^2\), the center is represented by the coordinates \((h, k)\).
When you look at the equation, \((x+4)^2 + (y+2)^2 = 20\), you can determine the circle’s center by identifying what \(h\) and \(k\) must be. In this equation, the expressions \(x+4\) and \(y+2\) correlate with \(x-h\) and \(y-k\) respectively. Therefore, \(h\) is \(-4\) and \(k\) is \(-2\). This reveals that the center of the circle is located at \((-4, -2)\).

• To find the center: Extract opposite values from the parentheses.
• Always note that \(x-h\) means the center's x-value is \(-h\).

Having this information allows you to better understand the geometry and position of the circle in a coordinate system.

Circle Radius Calculation

Calculating the radius is another unique aspect when determining the characteristics of a circle. The radius is the distance from the center to any point on the circle. In the equation \((x-h)^2 + (y-k)^2 = r^2\), \(r\) represents the radius.
In our example, given the equation \((x+4)^2 + (y+2)^2 = 20\), you can determine \(r^2\) equals 20. To find \(r\), you take the square root of 20, which results in \(r = \sqrt{20} = 2\sqrt{5}\).

• The number on the right side of the equation represents \(r^2\).
• To compute the radius, simply find the square root of that number.

This process gives you the length of the circle's radius, which is essential for further geometrical analysis.

Standard Form of a Circle Equation

The standard form of a circle equation provides a uniform method for representing circles algebraically. The standard form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the circle's center and \(r\) is the radius.
This form is neat and particularly useful because:

• Values \(h\) and \(k\) clearly define the center's location.
• \(r^2\) provides an immediate way to determine the radius' length.

For the equation \((x+4)^2 + (y+2)^2 = 20\), comparing it to the standard form allows you to extract \(h\) and \(k\) easily as \(-4\) and \(-2\) respectively. Furthermore, equating \(r^2 = 20\) helps deduce that \(r = 2\sqrt{5}\).
Thus, the standard form is a cornerstone for understanding and solving geometrical problems involving circles. It turns seemingly complex algebra into straightforward solutions, making life easier for students and mathematicians alike.