Problem 57 - Algebra and Trigonometry

Question

55–62 ? Find an equation of the circle that satisfies the given conditions. Center at the origin; passes through \((4,7)\)

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = 65\).

Step 1: Understand the Circle Equation

The general equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). In this case, the center is at the origin, so \(h = 0\) and \(k = 0\), leading to the equation \(x^2 + y^2 = r^2\).

Step 2: Insert the Given Point into the Equation

The given point is \((4, 7)\). Substitute \(x = 4\) and \(y = 7\) into the circle equation \(x^2 + y^2 = r^2\). This gives \(4^2 + 7^2 = r^2\).

Step 3: Solve for the Radius

Calculate \(4^2 + 7^2\) which is \(16 + 49 = 65\). So, \(r^2 = 65\).

Step 4: Write the Final Equation

Since we found that \(r^2 = 65\), the equation of the circle is \(x^2 + y^2 = 65\).

Key Concepts

Center of a Circle

The center of a circle is a crucial point as it defines the symmetry and balance of the geometric shape. In the equation for a circle, the center is denoted by the coordinates \((h, k)\). This point is special because it is equidistant from every point on the circle's boundary.
For our specific problem, the center of the circle is at the origin, meaning \((h, k) = (0, 0)\). When the center is at the origin, the circle is said to be in its simplest form. This simplification leads to a different form of the equation which does not include the variables \(h\) and \(k\).
Why does this matter? Because knowing the center of a circle helps us to accurately describe where the circle is located on the coordinate plane. When the center is at other coordinates, the general equation of a circle helps us track its exact position easily.

Radius of a Circle

The radius is another fundamental part of a circle. It is the distance from the center to any point on the circumference. In other words, it is the line segment between the center point and a point on the edge of the circle.
The radius can be found from the circle equation, typically represented in the form \((x - h)^2 + (y - k)^2 = r^2\). For circles centered at the origin, this simplifies to \(x^2 + y^2 = r^2\). In our exercise, after substituting the given point \((4, 7)\), we found the radius squared: \(r^2 = 65\).
The radius itself would be the square root of 65, but often we work with \(r^2\) to keep calculations simple. Understanding the radius is important because it affects the size of the circle. The larger the radius, the bigger the circle.

Standard Form of a Circle's Equation

The standard form of a circle's equation is an equation that represents all points \((x, y)\) lying on the circle. It takes the form of \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center, and \(r\) is the radius. This formula allows anyone reading it to immediately identify both the position and size of the circle.
In our exercise, since the center is at the origin (\((0, 0)\)), the equation simplifies to \(x^2 + y^2 = r^2\). This simpler formula is often easier to work with, especially in calculations or when graphing the circle.
By substituting the coordinates of any point on the circle into this equation, such as the point \((4, 7)\) in our example, we can find the radius squared, aiding in confirming or calculating the specifics of any circle. This standard form is a foundational concept in geometry and essential for various applications, from simple plotting to complex calculations.