Problem 76 - Algebra 2

Question

Write an equation of the circle with the given center and radius. Graph the circle. center \((1,1),\) radius 2

Step-by-Step Solution

Answer

The equation of the circle with center at \((1, 1)\) and radius 2 is \((x - 1)^2 + (y - 1)^2 = 4\). The graph should show a circle centered at \((1, 1)\) and touching the points \((1, 3)\), \((1, -1)\), \((3, 1)\), and \((-1, 1)\).

Step 1: Write the formula of a circle

The formula for a circle is: \((x - h)^2 + (y - k)^2 = r^2\)

Step 2: Substitute the given values

Substitute the given center \((h, k) = (1,1)\) and radius \(r = 2\) into the equation: \((x - 1)^2 + (y - 1)^2 = 2^2\)

Step 3: Simplify the equation

Simplify the equation to obtain: \((x - 1)^2 + (y - 1)^2 = 4\)

Step 4: Prepare to graph the circle

To graph the circle, apply the following guidelines: Plot the center at point \((1,1)\). From the center, measure 2 units up, down, left, and right, because the radius is 2. Draw a circle that passes through these four points.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric problems. It's an essential tool in mathematics that lets us analyze shapes and sizes in a plane using an ordered pair of numbers, typically represented as \(x, y\). This method merges geometry and algebra, allowing us to place geometric figures in a coordinate system and solve for various properties effectively.

In coordinate geometry, positioning of objects like points, lines, and circles are defined using coordinates. For a circle, the central part of coordinate geometry, the center is given by a point \(h, k\). This gives us a fixed position on the coordinate plane from which all properties of a circle are determined.

The crucial advantage of using coordinate geometry is how it simplifies complex geometric problems into algebraic expressions. By working with simple equations, like the one for a circle, it's possible to perform calculations, measure distances, and analyze relationships in a straightforward manner.

Equation of a Circle

The equation of a circle is a fundamental concept in both geometry and algebra. It's a special type of equation that describes all the points equidistant from a central point, known as the center of the circle. The standard form of a circle’s equation is:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \(h, k\) represents the coordinates of the circle's center, and \(r\) is the radius.

In our exercise, we used the given center \(1, 1\) and radius \(2\) to substitute into the standard equation. This gave us \((x - 1)^2 + (y - 1)^2 = 4\).

The excellent part about the circle’s equation is its ability to pop directly onto the coordinate plane. By knowing \(h, k\) and \(r\), anyone can visualize and verify the size and position of the circle directly from the mathematical formula. Simplifying the equation helps to understand how various parts contribute to the overall shape and make plotting it even easier.

Graphing Circles

Graphing circles is the visual representation of the circle's equation on a coordinate plane. To accurately graph a circle, follow these straightforward steps:

• Identify the circle’s center, \(h, k\). For example, \(1,1\) in our case.
• From the center, measure the radius in all four cardinal directions (up, down, left, right), using the value of \(r\); here it's \(2\).
• Plot each of these four points.
• Draw a smooth, uniform circle that passes through these points.

By understanding what each component of the circle’s equation represents, graphing becomes a straightforward task. The core idea is that every point on the circle is exactly \(r\) units away from the center. This uniform distance creates the perfectly round shape.

Graphing isn't just an exercise in representation; it solidifies our understanding of how mathematical principles translate to visual elements. It's also a practical skill that is useful in many areas, like physics, design, and more advanced mathematics. Remember, turning algebra into a picture not only clarifies concepts but also makes problem-solving more intuitive.