Problem 19 - Calculus Single Variable

Question

A circle is described in words. Give its Cartesian equation. The circle with radius 5 and center \((-4, \pi)\)

Step-by-Step Solution

Answer

The Cartesian equation is \((x + 4)^2 + (y - \pi)^2 = 25\).

Step 1: Understand the Standard Equation of a Circle

The standard equation of a circle in Cartesian coordinates 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 of the circle.

Step 2: Identify the Given Parameters

For the circle described in the problem, the center is \((-4, \pi)\) and the radius is 5. Therefore, \(h = -4\), \(k = \pi\), and \(r = 5\).

Step 3: Substitute the Values into the Standard Equation

Substitute the values \(h = -4\), \(k = \pi\), and \(r = 5\) into the standard equation: \[ (x - (-4))^2 + (y - \pi)^2 = 5^2 \] This simplifies to: \[ (x + 4)^2 + (y - \pi)^2 = 25 \]

Step 4: Write the Final Equation

The Cartesian equation of the circle is \[ (x + 4)^2 + (y - \pi)^2 = 25 \]

Key Concepts

Standard Equation of Circle

The standard equation of a circle in Cartesian coordinates is a fundamental building block when working with circles in mathematics. This equation helps describe the position and size of a circle on the Cartesian plane. It is given by the formula:
\[ (x - h)^2 + (y - k)^2 = r^2 \]where:

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

To use this equation, simply input the specific values for the center and the radius of your circle, and you'll have its equation ready to work with. This form is extremely handy because it allows quick transformations and adaptations to different circle problems through substitution of these parameters.
In our example, knowing the center \((-4, \pi)\) and the radius 5, we plug these values into the standard equation to derive the circle’s unique formula.

Circle Center

The circle's center is the fixed central point around which the circle is perfectly symmetrical. In the Cartesian plane, the center is given by a specific coordinate pair, \(h, k\), which becomes a crucial part of defining the circle's equation.
In the problem we are tackling, the center is given as \((-4, \pi)\). This means:

• \(h = -4\), indicating the horizontal shift from the origin.
• \(k = \pi\), showing the vertical shift.

These values tell us exactly where to plot the center of the circle when drawing it on a graph. It's essential to correctly identify these so they can be accurately substituted into the standard circle equation. This correct placement of the circle's center helps in revealing more about its location and potential intersections with other geometric figures.

Circle Radius

The radius of a circle is the distance from the center to any point on the boundary of the circle. It is a constant value, and knowing it helps determine the size of the circle. For calculating purposes, the radius \(r\) must be squared and plugged into the standard equation of the circle.
In our example, the radius provided is 5. Thus, \(r^2 = 25\). This squared value is used in the equation:
\[ (x + 4)^2 + (y - \pi)^2 = 25 \]Remember, the radius is always a positive number, representing a real physical distance on the Cartesian plane. It ensures that the circle is drawn to the right scale in relation to its center. Calculating the radius accurately is necessary for all subsequent procedures involving area and perimeter calculations of the circle.