Problem 6 - Calculus Single Variable

Question

The Cartesian equation of a circle is given. Sketch the circle and specify its center and radius. \((x+8)^{2}+(y-1)^{2}=16\)

Step-by-Step Solution

Answer

Center: (-8, 1); Radius: 4

Step 1: Identify the Circle Equation Form

The general equation of a circle is \[(x-h)^2 + (y-k)^2 = r^2\]where \((h, k)\) is the center of the circle and \(r\) is the radius. We will compare this general form with the given equation to identify the center and radius.

Step 2: Match with General Form

Given:\[(x+8)^2 + (y-1)^2 = 16\]Rewrite this as \((x - (-8))^2 + (y-1)^2 = 4^2\) so, the circle's center is \((-8, 1)\) and the radius is \(4\).

Step 3: Sketch the Circle

Draw a coordinate plane. Plot the center of the circle at the point \((-8, 1)\). From the center, measure a distance of 4 units in all directions to outline the circle. This completes the sketch of the circle.

Step 4: Find Center and Radius

The center of the circle is \((-8, 1)\) and the radius is \(4\). These values were identified by comparing the given circle equation with the general form.

Key Concepts

Cartesian coordinates

The Cartesian coordinate system is a two-dimensional plane defined by two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). It is used to pinpoint the location of points on a plane by means of two numbers, commonly referred to as coordinates. These coordinates are in the format \(x, y\), where \(x\) denoted the horizontal position and \(y\) denoted the vertical position.
The origin, where the axes intersect, is known as \(0, 0\). This setup is fundamental for graphing equations and shapes, such as circles, on a plane. By plotting points using their Cartesian coordinates, you can visualize mathematical relationships and functions.

• The positive direction of the x-axis moves rightward, and the positive direction of the y-axis moves upward.
• Negative values indicate movement in the opposite direction along the respective axis.

For circles, the Cartesian coordinate system helps not only in placing their centers but also in sketching these shapes accurately based on their equations.

center of a circle

In the context of a circle on the Cartesian coordinate plane, the center is a specific point from which all points on the circle are equidistant. The coordinates of the circle's center are given as \(h, k\) in the circle's equation \((x-h)^2 + (y-k)^2 = r^2\). Identifying the center is crucial because it offers a reference point to determine where the circle is positioned on the grid.
By comparing a given circle equation, such as \( (x+8)^2 + (y-1)^2 = 16\), to the standard form \( (x-h)^2 + (y-k)^2 = r^2\), you can determine that the center is located at \((-8, 1)\), stemming from the shifts in altering the \(x\) and \(y\) components of the equation:

• The value of \(h\) is the x-coordinate of the center, derived from \( (x - h)^2\). In the equation, \(+8\) becomes the negative shred of \(-8\).
• The value of \(k\) is the y-coordinate, represented as \( (y - 1)^2\).

Within the circle's context, the center provides the starting point from which the radius can be extended in all directions to draw the circle.

radius of a circle

The radius of a circle is a fundamental aspect defining its size, representing the distance from the circle's center to any point along its edge. In the Cartesian coordinate circle equation \( (x-h)^2 + (y-k)^2 = r^2\), \(r\) stands for the radius, and \(r^2\) is the completed square on the right side of the equation.
To find the radius, observe the equation's format and equate \(r^2\) to the constant noted at the end. For example, in \( (x+8)^2 + (y-1)^2 = 16\), this can be rewritten as \(r^2 = 16\), indicating a radius \(r\) of 4, since the square root of 16 is 4.

• The radius informs you how to draw the circle from its center, ensuring all points across its curve are exactly the radius distance away.
• Notably, the radius is always a positive value, reflecting a physical distance.

Understanding the radius is essential for drawing circles accurately on a Cartesian plane as it determines the circle's size and overall visual spread from its central point.