Find An Equation Of The Circle Whose Diameter Has Endpoints

Find an Equation of a Circle Given the Endpoints of its Diameter

Finding the equation of a circle given the endpoints of its diameter is a straightforward process using fundamental geometry and algebra. This article will guide you through the steps, providing clear explanations and examples to help you master this concept. This process involves finding the center and radius of the circle, which are essential components of the standard equation of a circle: (x - h)² + (y - k)² = r², where (h, k) represents the center and r represents the radius.

Understanding the Fundamentals

The diameter of a circle is a straight line segment that passes through the center and connects two points on the circle's circumference. Knowing the endpoints of the diameter allows us to determine both the center and the radius, which are crucial for constructing the circle's equation.

Step-by-Step Guide

Let's assume the endpoints of the diameter are A(x₁, y₁) and B(x₂, y₂). Here's how to find the equation of the circle:

1. Find the Center (h, k):

The center of the circle is the midpoint of the diameter. We can find the coordinates of the midpoint using the midpoint formula:

h = (x₁ + x₂) / 2 k = (y₁ + y₂) / 2

2. Find the Radius (r):

The radius is half the length of the diameter. We can find the length of the diameter using the distance formula between points A and B:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

The radius is then:

r = d / 2

3. Write the Equation:

Now that we have the center (h, k) and the radius (r), we can plug these values into the standard equation of a circle:

(x - h)² + (y - k)² = r²

Example:

Let's find the equation of a circle whose diameter has endpoints A(2, 4) and B(6, 0).

1. Finding the Center:

h = (2 + 6) / 2 = 4 k = (4 + 0) / 2 = 2 Center: (4, 2)

2. Finding the Radius:

d = √[(6 - 2)² + (0 - 4)²] = √(16 + 16) = √32 r = √32 / 2 = √8 = 2√2

3. Writing the Equation:

(x - 4)² + (y - 2)² = (2√2)² (x - 4)² + (y - 2)² = 8

Therefore, the equation of the circle is (x - 4)² + (y - 2)² = 8.

Common Mistakes to Avoid:

• Incorrect Midpoint Calculation: Double-check your arithmetic when calculating the midpoint of the diameter.
• Forgetting to Divide by 2: Remember that the radius is half the length of the diameter.
• Incorrect Squaring of the Radius: Ensure you correctly square the radius when substituting into the equation.

Advanced Applications:

This fundamental concept extends to more complex problems involving circles and their geometric properties. Understanding this process forms a crucial base for solving more advanced geometry problems and applications in other fields like coordinate geometry and calculus. Mastering this skill will significantly improve your understanding of circles and their equations.