Problem 8 - Calculus Single Variable

Question

The Cartesian equation of a circle is given. Sketch the circle and specify its center and radius. \(x^{2}+(y+7)^{2}=1\)

Step-by-Step Solution

Answer

The center of the circle is (0, -7) and the radius is 1.

Step 1: Identify the General Equation Format

The general equation for a circle in Cartesian coordinates is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Compare this with the given equation \(x^2 + (y + 7)^2 = 1\).

Step 2: Determine the Center of the Circle

From the equation \(x^2 + (y + 7)^2 = 1\), we can see that \(x\) is in the form \((x - h)^2\) with \(h = 0\) and \(y + 7\) can be rewritten as \((y - (-7))\). Thus, the center of the circle is \((0, -7)\).

Step 3: Find the Radius of the Circle

The right-hand side of the equation \(x^2 + (y + 7)^2 = 1\) represents \(r^2\). Therefore, \(r^2 = 1\) and the radius \(r\) is \(\sqrt{1} = 1\). Thus, the radius of the circle is 1.

Step 4: Sketch the Circle

To sketch the circle, plot the center at \((0, -7)\) on a Cartesian plane. Then, draw a circle around this center with a radius of 1. This means the circle will intersect the points \((1, -7)\), \((-1, -7)\), \((0, -6)\), and \((0, -8)\). Use these points as a guide to ensure the circle is properly drawn.

Key Concepts

Cartesian Coordinates

In order to understand the position and dimensions of a circle on a plane, it is vital to grasp the concept of Cartesian coordinates. These coordinates use two axes: horizontal (x-axis) and vertical (y-axis). Together, they form a grid on which we can locate any point in two-dimensional space by specifying an x-coordinate and a y-coordinate—often written as \((x, y)\). In the context of circles, the Cartesian coordinate system helps to precisely position the center of the circle and trace its outline by calculating points along its circumference. For instance, the circle from our exercise has its center at the point \((0, -7)\) based on the given equation.

Circle Radius

The radius of a circle is a crucial measurement—it represents the distance from the center of the circle to any point on its edge. This distance is the same all around the circle. In mathematical terms, the radius is derived from the circle's equation in the form \[(x - h)^2 + (y - k)^2 = r^2\]. Here, \(r\) stands for the radius. Thus, if you have an equation like \(x^2 + (y + 7)^2 = 1\), you directly identify \(r^2 = 1\), meaning the radius \(r\) is 1, because the square root of 1 is 1. Hence, this number tells us how wide the circle spreads from its center.

Graph Sketching

Graph sketching can be both an art and a science when it comes to visualizing circles. To sketch a circle from its equation, start by placing a dot at its center—using the Cartesian coordinates provided. For our example, you would place a dot at \((0, -7)\). Next, draw a circle around this point using the radius, which is 1 in this case. To make the circle accurate, you can plot a few key points. For a circle centered on \((0, -7)\) and having a radius of 1, this includes:

• \((1, -7)\)
• \((-1, -7)\)
• \((0, -6)\)
• \((0, -8)\)

These points touch the circumference of the circle. Join these points smoothly to complete your circle sketch. It's helpful to use a compass or a round object if you're doing this by hand.

General Equation Format

A general circle equation in Cartesian coordinates is essential for identifying all necessary components that define a circle. Typically written as \((x - h)^2 + (y - k)^2 = r^2\), each variable plays a key role:

• \((h, k)\): Represents the center of the circle in coordinates.
• \(r\): Represents the radius, determined by \(r^2\).

Recognizing this format helps in transforming any complex circle equation into a form that makes it easier to read and interpret. For example, our exercise begins with \(x^2 + (y + 7)^2 = 1\). By comparing it with the general form, we deduce:- Center at \((0, -7)\), extracted from noticing \(x - 0\) and \(y - (-7)\).- Radius is 1, since \(r^2 = 1\). Understanding this format simplifies solving circle-related problems significantly.