Problem 32 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at \((-0.7,-0.2) ;\) radius \(\sqrt{11}\)

Step-by-Step Solution

Answer

The circle's equation is \((x + 0.7)^2 + (y + 0.2)^2 = 11\).

Step 1: Understand the Standard Form of a Circle

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

Step 2: Identify the Circle's Center and Radius

From the problem, we know that the center \((h, k)\) of the circle is \((-0.7, -0.2)\) and the radius \(r\) is \(\sqrt{11}\).

Step 3: Insert the Center Coordinates into the Equation

Using the center point \((-0.7, -0.2)\), substitute \(h = -0.7\) and \(k = -0.2\) into the standard form equation to get \((x + 0.7)^2 + (y + 0.2)^2 = r^2\).

Step 4: Substitute the Radius into the Equation

Substitute \(r = \sqrt{11}\) into the equation. Remember that \(r^2 = (\sqrt{11})^2 = 11\). This gives us \((x + 0.7)^2 + (y + 0.2)^2 = 11\).

Key Concepts

Center of a Circle

The center of a circle is a crucial part of understanding its equation. In coordinate geometry, we often think of a circle as being perfectly symmetrical around a central point, known as the center. It's a point on the plane from which all points that form the circle are equidistant. This distance is the radius of the circle.

In mathematical form, if the center of the circle is \(h, k\), we use these coordinates when writing the equation. For example, if a circle is centered at \((-0.7, -0.2)\), you plug these values into the standard form equation of a circle \((x - h)^2 + (y - k)^2 = r^2\)\.

• \(h\) and \(k\) represent the x and y coordinates of the circle's center.
• The circle is symmetric around this point.

Radius of a Circle

The radius of a circle is the constant distance from its center to any point on its perimeter. Imagine stretching a string from the center to the edge; the length of the string represents the radius.

In our given exercise, the radius is \(\sqrt{11}\). To use this in the equation, it's crucial to remember the square of the radius is needed. In mathematical terms, \((\sqrt{11})^2 = 11\), which simplifies your substitute into the equation.

• The radius defines the size of the circle.
• It is always a positive number.
• Knowing the radius helps complete the circle's equation.

Standard Form of a Circle

Writing the equation of a circle in its standard form helps clearly define its geometric properties. This equation is denoted by \( (x - h)^2 + (y - k)^2 = r^2 \). Here’s how you interpret it:

- Each term represents distances squared, centered around \(h\) and \(k\).- By arranging the circle equation in this format, it visually communicates the circle's size and position on the coordinate plane.

Using the given example, for a circle with center \((-0.7, -0.2)\) and radius \(\sqrt{11}\), our equation becomes \((x + 0.7)^2 + (y + 0.2)^2 = 11\). This demonstrates both the circle's location and how far it extends from the center in all directions.

• Ensures consistency in how we describe circles.
• Makes calculations and graphing simpler and clearer.
• Helps solve problems involving circle intersections and tangents.