Problem 75 - Algebra 2

Question

Write an equation of the circle with the given center and radius. Graph the circle. center \((-3,1),\) radius 5

Step-by-Step Solution

Answer

The equation of the circle is \( (x+3)^2 + (y-1)^2 = 25 \).

Step 1: Identify the center and the radius of the circle

The center of the circle, \((h,k)\), is given as \((-3,1)\), and the radius, r, is given as 5.

Step 2: Substitute the center and the radius into the standard circle equation

Substitute \((h,k) = (-3,1)\) and \(r = 5\) into the standard circle equation, \( (x-h)^2 + (y-k)^2 = r^2 \). This gives the equation of the circle as \( (x+3)^2 + (y-1)^2 = 25 \).

Step 3: Graph the circle using the equation

The circle should be drawn with its center at (-3, 1) and with a radius of 5. All points that are a distance of 5 units away from the center \((-3,1)\) should be on the circle.

Key Concepts

Graphing Circles

When graphing a circle, the most crucial elements are its center and radius. The center (h, k) provides the exact location where the circle is pinned on the coordinate plane, while the radius r tells you how far out the circumference stretches.
Imagine a pin at the center with a string attached. The radius is the fixed length of the string, dictating how far you can stretch it out to draw the circle.

• Center The center in our example is located at (-3, 1). You can easily plot this point as a reference on your graph.
• Radius A radius of 5 means that every point on the circle is exactly 5 units away from (-3, 1).

Begin at the center and count 5 units outward in all directions. Mark these points on the graph and connect them smoothly to form your circle. Remember, your circle should look even and symmetrical around its center.

Coordinate Geometry

Coordinate Geometry helps us understand the relationships between shapes using their numerical coordinates. With a system that links algebra and geometry, solving problems involving shapes like circles becomes more straightforward.
This area of math allows you to make sense of where points are located and how they relate to each other on a graph.

• Coordinates Coordinates like (-3, 1) describe a specific point on the plane. The x-coordinate shows the horizontal position, while the y-coordinate shows the vertical one.
• Distance The concept of distance is key when it comes to circles. The formula for a circle uses distance to keep each point on its circumference an equal distance from the center.

Understanding these basics is essential. It sets the foundation for graphing and interpreting more complex geometric figures.

Standard form of Circle Equation

The standard form of a circle's equation is a tool that helps you describe the circle algebraically. It takes into account both the center and the radius to give you a clear representation of the circle's properties.
This equation is \((x-h)^2 + (y-k)^2 = r^2\), where (h, k) represents the center and r is the radius.

• Substitution Plugging in the values for h, k, and r transforms the equation into a form specific to the circle in question. For our example, substituting h = -3, k = 1, and r = 5 yields \((x+3)^2 + (y-1)^2 = 25\).
• Visual Interpretation This equation tells you the distance squared from any point (x, y) to the center must be precisely 25. It encodes the rules you need to graph the circle accurately.

By mastering this form, you gain insight into both construction and analysis, simplifying how you work with circles in geometry.