Problem 49 - Algebra and Trigonometry

Question

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

The circle is centered at (0,0) with a radius of 3.

Step 1: Recognize the Equation of a Circle

The equation given is \(x^2 + y^2 = 9\). This represents a circle in standard form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Identify the Center of the Circle

For the equation \(x^2 + y^2 = 9\), you can see that there are no additional terms "+kx" or "+ky", and hence both \(h\) and \(k\) are zero. Therefore, the center of the circle is \((0,0)\).

Step 3: Calculate the Radius of the Circle

The equation \(x^2 + y^2 = 9\) can be compared to the standard form to identify the radius. Since the right side of the equation is \(9\), which corresponds to \(r^2\) in the standard form, the radius \(r\) is \(\sqrt{9}=3\).

Step 4: Sketch the Graph

To sketch the graph, plot the center of the circle at the origin \((0,0)\). Then, draw a circle around the center with a radius of 3. It will pass through the points (3,0), (-3,0), (0,3), and (0,-3) on the axes.

Key Concepts

Center of the Circle

The center of a circle in an equation provides the starting point for describing the circle's location on a coordinate plane. When you look at the equation of a circle in its standard form, \[(x-h)^2 + (y-k)^2 = r^2,\] the center of the circle is given by the coordinates \(h, k\). In simpler terms, if you imagine placing the circle on a plain sheet of graph paper, \(h, k\) represents the spot where the tip of a pencil would rest to draw the circle from the inside out. For example, with the equation \[x^2 + y^2 = 9,\]since there are no \(x\) or \(y\) terms added or subtracted, it implies \(h = 0\) and \(k = 0\). This means the center of the circle is at the origin, \(0,0\). It's the most basic form, with the origin being the default center unless shifted otherwise in the equation.

Radius of the Circle

The radius of a circle links the center to any point along the curved line that forms the circle's border. Think of it like a string stretched from the middle to the outer edge of the circle. The standard equation for a circle shows this as \(r\), where \(r^2\) is written as the number on the other side of the equation compared to the parts with \(x\) and \(y\). In our specific problem of \[x^2 + y^2 = 9,\] the number \(9\) takes the place of \(r^2\). To find \(r\), we simply take the square root: \[\begin{aligned}r = \sqrt{9} = 3.\end{aligned}\] This 3-unit distance tells us how far any point on the circle's edge is from the center, \(0,0\). This constant distance forms the circle's perfectly rounded shape.

Standard Form of a Circle Equation

Understanding the standard form of a circle equation is crucial for easily identifying the circle's key characteristics like its center and radius. The standard form of a circle's equation is written as: \[(x-h)^2 + (y-k)^2 = r^2\]where:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

If the equation of the circle is given similarly to our example \[x^2 + y^2 = 9,\]it indicates a simplified standard form with the center at the origin (\((0,0)\)) and the radius squared \(r^2\) on the right-hand side. Translating from this format to understand the circle's properties is straightforward when the terms match the \[(x-h)^2 + (y-k)^2 = r^2\]placement. Always look for a match in structure to identify centers and radii quickly!