Problem 56 - Algebra and Trigonometry

Question

55–62 ? Find an equation of the circle that satisfies the given conditions. Center \((-1,-4) ; \quad\) radius 8

Step-by-Step Solution

Answer

The equation of the circle is \((x + 1)^2 + (y + 4)^2 = 64\).

Step 1: Identify the Center and Radius

The center of the circle is given as \((-1, -4)\) and the radius is 8.

Step 2: Recall the Standard Equation of a Circle

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

Step 3: Substitute Given Values into Equation

Substitute \(h = -1\), \(k = -4\), and \(r = 8\) into the equation: \((x + 1)^2 + (y + 4)^2 = 8^2\).

Step 4: Simplify the Equation

Compute \(8^2\): \((x + 1)^2 + (y + 4)^2 = 64\).

Key Concepts

Standard Equation of a Circle

The standard equation of a circle is an essential tool in understanding circle geometry. It provides a simple way to describe a circle using algebraic expressions. A circle is a round shape with all points equidistant from the center. In mathematical terms, this uniform distance is known as the radius.

The standard equation of a circle is written as:\[(x - h)^2 + (y - k)^2 = r^2\]In this equation:

• \((h, k)\) represents the coordinates of the center of the circle.
• \(r\) denotes the radius of the circle.

This formula is intriguing because it encompasses all circular properties in a simple algebraic expression. To use it correctly, one substitutes the specific values of the center coordinates and the radius into the placeholders \(h\), \(k\), and \(r\). This way, you can describe any circle concisely and efficiently.

Center of a Circle

The center of a circle is a crucial point from which all other points on the circle are equally distant. This central location determines the position of the circle on the coordinate plane. Whether you want to draw or calculate the properties of a circle, knowing its center is vital.

When you look at the standard circle equation \((x - h)^2 + (y - k)^2 = r^2\), the variables \(h\) and \(k\) specify the center's coordinates. Specifically:

• \(h\) is the x-coordinate of the circle's center.
• \(k\) is the y-coordinate of the circle's center.

For example, if the center is given as \((-1, -4)\), it means that from the origin, the center is one unit to the left and four units down. Having clear coordinates allows you to plot the circle and understand how it's positioned in the Cartesian plane.

Radius of a Circle

The radius is a straight line from the center of a circle to any point on its circumference. It is one of the defining properties of a circle, describing how "big" or "small" the circle is.

The radius is represented by the letter \(r\) in the standard equation of a circle, \((x - h)^2 + (y - k)^2 = r^2\). Here are some key points about the radius:

• The radius is always a positive value. It measures the distance, so it must be more than zero.
• Squared in the equation, the radius helps maintain the formula's balance. When solving or transposing the equation, don't forget to deal with this squared component.
• For practical purposes like drawing or calculation, knowing the radius allows you to construct the circle accurately.

In the example where the radius is 8, you would substitute 8 in place of \(r\) in the circle equation, simplifying it to determine the full geometric definition of the circle.