Problem 83 - Intermediate Algebra

Question

Find the center and radius of each circle. $$ \text { a. }(x-4)^{2}+(y+7)^{2}=28 \quad \text { b. }(x+4)^{2}+(y-7)^{2}=28 $$

Step-by-Step Solution

Answer

Circle a: Center (4, -7), Radius \(2\sqrt{7}\); Circle b: Center (-4, 7), Radius \(2\sqrt{7}\).

Step 1: Identify the Standard Equation of a Circle

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

Step 2: Identify the Center and Radius for Circle a

For the equation \((x - 4)^2 + (y + 7)^2 = 28\), compare it with the standard form. This gives \(h = 4\), \(k = -7\). Therefore, the center is \((4, -7)\). To find the radius, solve \(r^2 = 28\), giving \(r = \sqrt{28} = 2\sqrt{7}\).

Step 3: Identify the Center and Radius for Circle b

For the equation \((x + 4)^2 + (y - 7)^2 = 28\), compare it with the standard form. This gives \(h = -4\), \(k = 7\). Therefore, the center is \((-4, 7)\). To find the radius, solve \(r^2 = 28\), giving \(r = \sqrt{28} = 2\sqrt{7}\).

Key Concepts

Center of a Circle

The center of a circle is a fundamental concept when studying the geometry of circles. It is defined as the exact middle point from which every point on the circle is equidistant.
This point is often represented by the coordinates \((h, k)\). To find the center of a circle when given a circle's equation in the form \((x-h)^2 + (y-k)^2 = r^2\), identify \(h\) and \(k\) directly from the equation.

• In the expression \((x-h)^2\), \(h\) represents the x-coordinate of the center. If expressed as \((x-4)^2\), for example, \(h\) is \(4\).
• For \((y-k)^2\), \(k\) represents the y-coordinate. If shown as \((y+7)^2\), then \(k\) is \(-7\), because the formula uses subtraction to isolate \(k\).

Thus, in equation "a" \((x - 4)^2 + (y + 7)^2 = 28\), the center is \((4, -7)\), while in equation "b" \((x + 4)^2 + (y - 7)^2 = 28\), the center is \((-4, 7)\). These are the reference points from which radius measurements radiate outward.

Radius of a Circle

The radius of a circle is a straight line drawn from the center of the circle to any point on its outer edge. Understanding the radius helps define the size of the circle as it represents half the circle's diameter. In mathematical terms, the radius \(r\) can be found using the formula \((x-h)^2 + (y-k)^2 = r^2\). To extract \(r\), simply take the square root of the value on the right side of the equation!

• Consider the expression \(r^2 = 28\). To find the radius, calculate \(r = \sqrt{28}\), which simplifies to \(r = 2\sqrt{7}\).

Both circle "a" and circle "b" in the given problem share the same radius — \(2\sqrt{7}\).
By calculating the radius, you determine how far each side of the circle extends from the center, which is crucial for understanding the circle's dimensions.

Standard Form of a Circle

The standard form of a circle is a way of expressing a circle's equation that clearly outlines its center and radius. This form is given by the equation \((x-h)^2 + (y-k)^2 = r^2\).

• Here, \((h, k)\) represents the center of the circle.
• The term \(r^2\) indicates the radius squared.

By rewriting any general circle equation into this standard form, students can quickly identify both the center and the radius of the circle.
This method is particularly useful when dealing with exercises that involve finding properties of circles just by observation and manipulation of algebraic expressions. In the provided exercises:

• Equation "a": \((x - 4)^2 + (y + 7)^2 = 28\) is in standard form with \(h = 4\), \(k = -7\), and \(r^2 = 28\).
• Equation "b": \((x + 4)^2 + (y - 7)^2 = 28\) follows the same pattern with \(h = -4\), \(k = 7\), and a similar \(r^2\) value.

Converting to and understanding this form ensures that any circle's attributes can be readily accessed.