Problem 14 - Beginning and Intermediate Algebra

Question

Identify the center and radius of each circle and graph. $$x^{2}+y^{2}=25$$

Step-by-Step Solution

Answer

The center of the circle is at the origin (0,0) and its radius is 5. To graph the circle, mark the center on the coordinate plane, measure a distance of 5 units along the x and y-axes, and draw a circle connecting the points.

Step 1: Identify the center and radius of the circle

Compare the given equation \(x^2+y^2=25\) with the general equation of a circle \((x-h)^2+(y-k)^2=r^2\). The given equation can be written as \((x-0)^2+(y-0)^2=5^2\). So, we have \(h=0\), \(k=0\) and \(r=5\). Thus, the center of the circle is at the origin (0,0) and its radius is 5.

Step 2: Graph the circle

To graph the given circle with center (0,0) and radius 5, follow these steps: 1. Mark the center of the circle on the coordinate plane. 2. From the center, measure a distance equal to the radius (5 units in this case) along the x and y-axes. 3. Draw the circle by connecting the points at a distance of 5 units from the center. The circle with center (0,0) and radius 5 has been graphed.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric figures. It translates visual geometric shapes, like circles or lines, into algebraic equations which can be analyzed mathematically. This approach allows us to utilize coordinate systems, such as the Cartesian plane, to understand and solve geometric problems.

• The Cartesian plane consists of two perpendicular axes: the x-axis and the y-axis.
• Each point on the plane is identified by a pair of coordinates \( (x, y) \).
• Understanding coordinate geometry helps in representing equations and graphing geometric figures.

Center of a Circle

The center of a circle in coordinate geometry is a critical point that determines the circle's position on the plane. The standard equation for a circle is \[(x-h)^2+(y-k)^2=r^2,\]where \((h, k)\) is the center of the circle.

• In this formula, \( h \) and \( k \) are the coordinates of the circle's center.
• For the equation \(x^2 + y^2 = 25\), the center is \( (0, 0) \), meaning the circle is centered at the origin.
• Recognizing the center helps in graphing and analyzing the circle's properties.

Radius

The radius of a circle is the constant distance from its center to any point on its circumference. In coordinate geometry, the radius is derived from the circle's equation. For a circle represented by \[(x-h)^2+(y-k)^2=r^2,\]\(r\) is the radius. It is a non-negative number defining the size of the circle.

• For the given equation \(x^2 + y^2 = 25\), we rewrite it as \( (x-0)^2 + (y-0)^2 = 5^2\).
• Thus, the radius \( r = 5 \).
• Knowing the radius helps in drawing the circle accurately.

Graphing Circles

Graphing circles on the coordinate plane involves plotting the circle using its center and radius. This visual representation helps to better understand the circle's position and size.

• Begin by marking the center on the Cartesian plane.
• Using the radius, measure equal distances (in all directions) from the center to plot points.
• Connect these points to form the circle, ensuring all points are equidistant from the center.

In our example, the center is at \( (0,0) \) with a radius of 5. You can draw the circle by maintaining this consistent distance of 5 units from the center all around.