Problem 74 - Algebra 2

Question

Write an equation of the circle with the given center and radius. Graph the circle. center \((0,0),\) radius 3

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = 9\). The graph is a circle centred at the origin (0,0) with a radius of 3 units.

Step 1: Formulate the Circle Equation

Since the center is at the origin, h and k both are 0. The radius of the circle is given as 3, so r is 3. Substituting these values into the standard circle equation gives \((x-0)^2 + (y-0)^2 = (3)^2\), or, simplifying, \(x^2 + y^2 = 9\). This is the equation of the circle.

Step 2: Graph the Circle

Plot the center of the circle at the origin, (0,0). Draw a circle with radius of 3 units. The circle passes through the points (3,0), (0,3), (-3,0), and (0,-3), on the axes. Double check that the circle appears accurate and touches these points.

Key Concepts

Equation of a Circle

To understand the equation of a circle in coordinate geometry, you must first know what a circle is. A circle includes all points that are a fixed distance from a central point, known as the center. This fixed distance is called the radius.

In its standard form, the equation of a circle with center ewline at point ewline (h, k) and radius ewline r is written as:\[ (x - h)^2 + (y - k)^2 = r^2 \]For example, if the center of the circle is \((0,0)\) and the radius is 3, the equation becomes:\[ (x - 0)^2 + (y - 0)^2 = 3^2 \]Simplifying, this gives:\[ x^2 + y^2 = 9 \]Each term of the equation helps describe the distance relationship from the center of the circle to its perimeter.

Graphing Circles

Graphing a circle on the coordinate plane is straightforward once you have the equation. Start with the circle's center as your reference point, which will be plotted on the graph. In the example with a center at \((0,0)\) and a radius of 3:

• Start by placing a point at the origin \((0,0)\), the center.
• From this center, count 3 units in all four directions: up, down, left, and right along the x-axis and y-axis.
• Place points at \((3,0)\), \((-3,0)\), \((0,3)\), and \((0,-3)\).

Once these boundary points are established, sketch the circle by drawing a smooth curve that connects these points evenly around the center.
This visual representation ensures the circle is correctly graphed, especially when it passes through the key points on the axes corresponding to its radius.

Standard Circle Equation

Understanding the standard form of a circle's equation is central to solving many circle-related problems. This equation not only helps define the location and size of a circle but also assists in graphing and analyzing circle properties.
The formula,\[ (x - h)^2 + (y - k)^2 = r^2 \]reveals vital details:

• \(h\) and \(k\) represent the x and y coordinates of the circle's center.
• \(r\) is the circle's radius.
• \((x, y)\) are the coordinates of any point on the circle.

The whole equation signifies that any point \((x, y)\) lies exactly a distance \(r\) away from the center of the circle. Learning how to apply this equation is important for both mathematical applications and real-world circle descriptions.

Radius of a Circle

The radius is a key element in the equation and the graph of a circle. This is because the radius determines the circle's size and how it spreads across the coordinate plane.
It is always the distance between the center and any point on the circle's perimeter.
In the equation\[ (x - h)^2 + (y - k)^2 = r^2 \],the radius \(r\) appears as \(r^2\). So, to find the radius, you must take the square root of the number on the right side of the equation.
For instance, when given the equation \(x^2 + y^2 = 9\), the radius \(r\) is \(3\), since \(r^2 = 9\) and the square root of 9 is 3.
This simple computation is crucial as it enables you to sketch the circle accurately, ensuring its size is correctly represented on paper or digital screen.