Problem 47 - Intermediate Algebra

Question

Write an equation of the circle with the given center and radius. $$(-5,4) ; 3 \sqrt{5}$$

Step-by-Step Solution

Answer

The equation is \((x + 5)^2 + (y - 4)^2 = 45\).

Step 1: Understand the Standard Equation of a Circle

The standard form of the equation for a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). This formula is essential to writing the equation of a circle when given the center and radius.

Step 2: Substitute the Center Coordinates

Insert the center coordinates \((-5, 4)\) into the formula. This means you will substitute \(h = -5\) and \(k = 4\) into the equation, transforming it to \((x + 5)^2 + (y - 4)^2 = r^2\).

Step 3: Substitute the Radius

Substitute the given radius \(3 \sqrt{5}\) into the equation. Since \(r^2\) is needed, square the radius: \((3 \sqrt{5})^2 = 9 \times 5 = 45\). Now the equation becomes \((x + 5)^2 + (y - 4)^2 = 45\).

Step 4: Finalize the Equation of the Circle

After substituting both the center coordinates and radius, the final equation of the circle is \((x + 5)^2 + (y - 4)^2 = 45\). This represents the equation of a circle with center at \((-5, 4)\) and radius \(3 \sqrt{5}\).

Key Concepts

Standard Form

The standard form is a simple yet powerful way to represent the equation of a circle. It allows us to express the circle's properties using a mathematical equation. The formula is: \[(x - h)^2 + (y - k)^2 = r^2\] Here,

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

Using standard form makes it straightforward to visualize where a circle lies on the coordinate plane and how wide it spans. Knowing the standard form is the cornerstone to solving problems related to finding a circle's equation.

Center of a Circle

The coordinates \((h, k)\) mark the exact spot in the plane that represents the center of the circle. For any given circle, the center is vital as it's the point from which all points on the circle maintain a constant distance, the radius. In the given exercise, the center is

• \(h = -5\)
• \(k = 4\)

This locates our circle in the fourth quadrant of the coordinate plane, to the left and above the origin. Identifying this central point is crucial for inserting the correct values into the standard form equation.

Radius

The radius \(r\) of a circle is the constant distance from the center to any point on the circle. In the exercise, the radius is given as \(3 \sqrt{5}\). To incorporate the radius in the standard circle equation, we need to square this value. This process looks like this:

• First, recognize \(r = 3 \sqrt{5}\)
• Calculate the square: \((3 \sqrt{5})^2 = 9 \times 5 = 45\)

Squaring the radius is necessary as the standard equation uses \(r^2\) to translate the radius into mathematical terms accurately.

Substitute Values

Substituting values is the step where we plug our known center and radius into the standard form equation to shape the circle's equation. Consider it like painting by numbers, plugging in values to complete the picture. By substituting the

• center \((-5, 4)\) and
• the calculated \(r^2\)

into the equation, we derive \[(x + 5)^2 + (y - 4)^2 = 45\] This transformation solidifies the abstract formula into a concrete equation that represents our specific circle. It's essential to perform these substitutions accurately to ensure the equation correctly embodies the circle's specified properties.