Problem 17 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((4,1) ;\) radius \(=5\)

Step-by-Step Solution

Answer

The equation of the circle with center \((4, 1)\) and radius \(5\) is \((x - 4)^2 + (y - 1)^2 = 25\).

Step 1: Substitute the center and the radius

Substitute the given center point \((4, 1)\) and the radius \(5\) into the standard equation of the circle: \((x - 4)^2 + (y - 1)^2 = 5^2\)

Step 2: Simplify the equation

Simplify the equation: \((x - 4)^2 + (y - 1)^2 = 25\) That's it! The equation of the circle with center \((4, 1)\) and radius \(5\) is: \((x - 4)^2 + (y - 1)^2 = 25\)

Key Concepts

Center of a Circle

The center of a circle is a fundamental concept that you should master when learning about circles in geometry. The center is a fixed point around which every point on the circle is equidistant. In simpler terms, it is the point that is exactly in the middle of the circle. Knowing the center is crucial, as it helps define the position of the circle in a coordinate plane.

Suppose we have the circle's center described by the coordinates \(h, k\). This means that \(h\) represents the horizontal (x-coordinate) and \(k\) represents the vertical (y-coordinate) position of the center point.

For the provided problem, the center of the circle is \( (4, 1) \). Here, 4 is the x-coordinate, and 1 is the y-coordinate. This center point tells us exactly where the circle is located on the grid. This information is essential since it directly feeds into the standard equation of the circle.

Radius of a Circle

The radius of a circle is another essential aspect you need to understand. It represents the distance from the center of the circle to any point on the circle. This distance remains constant no matter where on the circle you measure from. Think of the radius as the length of a spoke of a bicycle wheel, stretching from the hub (center) to the wheel's rim.In mathematical terms, the radius is expressed as \(r\). Every circle's equation incorporates this dimension, ensuring you grasp how large the circle is. For example, if a circle has a radius of 5, this means every point on the circle is exactly 5 units away from the center, \( (4, 1) \). Important points to remember about the radius include:

• Determines the size of the circle.
• Is always a non-negative number.
• Takes part in the circle's equation as the square of its value.

By measuring or knowing the radius, you can visualize the circle's size and curvature.

Standard Equation of a Circle

The standard equation of a circle is a mathematical representation that defines all points on a circle in relation to its center and radius. This equation is crucial because it gives a precise way to describe any circle on a coordinate plane. The general form is:\[(x - h)^2 + (y - k)^2 = r^2\]Here:

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

The equation essentially states that the distance of any point \( (x, y) \) from the center \( (h, k) \) matches the radius.To find the specific equation for a circle with a center at \( (4, 1) \) and a radius of 5, substitute these values into the standard form:\[(x - 4)^2 + (y - 1)^2 = 5^2\]Simplifying gives:\[(x - 4)^2 + (y - 1)^2 = 25\]The presence of the squared terms ensures the circle's geometric precision. Each variable's role helps distinctly define the circle's size and location on a graph. By understanding this equation, you'll be able to map and manipulate circles easily in mathematical problems.