Problem 36 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$\text { Center }(-3,5), r=3$$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center (-3,5) and radius 3 is \((x+3)^2 + (y-5)^2 = 9\).

Step 1: Identify and Note Circle Parameters

To begin, identify the circle's parameters given in the exercise. The center is (-3,5) and the radius is 3.

Step 2: Write down Standard Form of the Equation of a Circle

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

Step 3: Substitute Circle Parameters into the Standard Form

We substitute the given values into the equation \((x - (-3))^2 + (y - 5)^2 = 3^2\). Then simplify this to \( (x+3)^2 + (y - 5)^2 = 9\).

Key Concepts

Standard Form

The standard form of a circle equation is an elegant way to describe every point that forms a circle on a Cartesian coordinate system. Here is the formula:\[ (x - h)^2 + (y - k)^2 = r^2 \]This equation makes it quite simple to visualize and understand the properties of a circle because it incorporates both the center \(h, k\) and the radius \(r\) directly into the equation. The terms \(x\) and \(y\) are variables representing all the points on the circle.

• In \( (x - h)^2 \), \(h\) shifts the circle horizontally.
• In \( (y - k)^2 \), \(k\) shifts the circle vertically.
• \(r^2\) provides the circle's radius squared, a measure of its size.

Using this standardized format makes it easier to quickly determine a circle's geometry from an equation.

Circle Parameters

Circle parameters are vital to fully comprehend the circle's characteristics. These parameters include:

• Center: Represented by the coordinates \( (h, k) \), the center is the point from which all points on the circle are equidistant.
• Radius: denoted as \( r \), is the distance from the center to any point on the circle. It defines the size of the circle.

Knowing these parameters allows you to plug values directly into the standard form equation. For example, if the center is \(-3, 5\), you'd use these for \(h\) and \(k\) respectively. If the radius is \(3\), then in the equation, you would substitute \( r \) with \(3\). This way, the circle parameters provide a straightforward path to determine every relevant detail about a circle.

Radius

The radius is one of the most critical parameters in defining a circle. It is always a positive real number and is consistent throughout the circle. The radius is calculated from the distance formula if not given directly. If the center of a circle is \(h, k\) and any point on the circle is \(x, y\), you could use:\[ r = \sqrt{(x-h)^2 + (y-k)^2} \]The equation in standard form eventually squares the radius, shown as \( r^2 \) to simplify calculations. For instance, when given a radius of \(3\), squaring gives \(9\), directly influencing the circle's equation formation. Thus, understanding the radius's role is crucial in determining a circle's size and interpreting the equation correctly.

Center (h,k)

The center point of a circle is crucial since it serves as the circle's anchor. Denoted as \(h, k\), it signifies the circle's fixed point from which the radius extends in all directions.This point is vital when converting an equation into its standard form. In the formula \[ (x-h)^2 + (y-k)^2 = r^2 \]

• \(h\) determines the horizontal shift of the circle on the \(x\)-axis.
• \(k\) determines the vertical shift on the \(y\)-axis.

Given a center of \(-3, 5\), we plug \(-3\) and \(5\) into \(h\) and \(k\) respectively. Therefore, the equation properly pivots the circle's placement in the coordinate plane. Understanding \(h, k\) helps embrace the geometric role of the center in circle equations clearly.