Homework 8 Equations Of Circles Answers

Homework: 8 Equations of Circles — Answers and Comprehensive Guide

Homework assignments on equations of circles can be challenging, but mastering them is crucial for understanding more advanced concepts in geometry and algebra. This comprehensive guide provides detailed answers and explanations for eight sample equations of circles, covering various scenarios and problem-solving techniques. We’ll delve into the standard form, general form, and how to derive essential information such as the center and radius from the equation. Let's tackle these problems and solidify your understanding!

Understanding the Equation of a Circle

Before we dive into the specific problems, let's refresh our understanding of the equation of a circle. The standard form of the equation of a circle is:

(x - h)² + (y - k)² = r²

Where:

• (h, k) represents the coordinates of the center of the circle.
• r represents the radius of the circle.

The general form of the equation of a circle is:

x² + y² + Dx + Ey + F = 0

While less intuitive, the general form can be converted to the standard form through a process called completing the square. This allows us to easily identify the center and radius.

Eight Sample Problems and Solutions

Let's work through eight example problems, showcasing different variations and challenges you might encounter in your homework.

Problem 1: Find the center and radius of the circle with equation (x - 3)² + (y + 2)² = 16.

Solution: This equation is already in standard form. By comparing it to the standard equation (x - h)² + (y - k)² = r², we can directly identify:

• Center (h, k) = (3, -2) (Remember that the signs are reversed in the equation).
• Radius r = √16 = 4

Problem 2: Write the equation of a circle with center (0, 0) and radius 5.

Solution: Using the standard form and plugging in the given values:

• (x - 0)² + (y - 0)² = 5²
• x² + y² = 25

Problem 3: Find the center and radius of the circle with equation x² + y² - 6x + 4y - 12 = 0.

Solution: This is in general form. We need to complete the square to convert it to standard form:

1. Group x and y terms: (x² - 6x) + (y² + 4y) - 12 = 0
2. Complete the square for x terms: Take half of the coefficient of x (-6), square it ((-3)² = 9), and add and subtract it: (x² - 6x + 9 - 9)
3. Complete the square for y terms: Take half of the coefficient of y (4), square it ((2)² = 4), and add and subtract it: (y² + 4y + 4 - 4)
4. Rewrite the equation: (x² - 6x + 9) + (y² + 4y + 4) - 9 - 4 - 12 = 0
5. Simplify: (x - 3)² + (y + 2)² = 25
6. Identify center and radius: Center (h, k) = (3, -2); Radius r = √25 = 5

Problem 4: Write the equation of a circle with center (-1, 3) and a radius of 2.

Solution: Using the standard form:

• (x - (-1))² + (y - 3)² = 2²
• (x + 1)² + (y - 3)² = 4

Problem 5: Find the center and radius of the circle with equation x² + y² + 8x - 10y + 16 = 0.

Solution: Completing the square:

1. Group x and y terms: (x² + 8x) + (y² - 10y) + 16 = 0
2. Complete the square: (x² + 8x + 16 - 16) + (y² - 10y + 25 - 25) + 16 = 0
3. Simplify: (x + 4)² + (y - 5)² = 25
4. Identify center and radius: Center (h, k) = (-4, 5); Radius r = √25 = 5

Problem 6: Write the equation of the circle passing through the point (4, 2) and having its center at (1, -1).

Solution: First, we need to find the radius. The distance between the center (1, -1) and the point (4, 2) is the radius:

1. Use the distance formula: r = √[(4 - 1)² + (2 - (-1))²] = √(9 + 9) = √18
2. Write the equation: (x - 1)² + (y + 1)² = 18

Problem 7: Find the center and radius of the circle with equation 2x² + 2y² - 12x + 8y - 24 = 0.

Solution: Divide the entire equation by 2 to get the standard form coefficients:

1. Divide by 2: x² + y² - 6x + 4y - 12 = 0
2. Complete the square (same as Problem 3): (x - 3)² + (y + 2)² = 25
3. Identify center and radius: Center (h, k) = (3, -2); Radius r = 5

Problem 8: A circle has a diameter with endpoints at (-2, 1) and (4, 3). Find the equation of the circle.

Solution:

1. Find the center: The midpoint of the diameter is the center. Midpoint = ((-2 + 4)/2, (1 + 3)/2) = (1, 2)
2. Find the radius: The distance between the center (1, 2) and either endpoint is the radius. Using the distance formula with (4, 3): r = √[(4 - 1)² + (3 - 2)²] = √10
3. Write the equation: (x - 1)² + (y - 2)² = 10

Further Exploration and Practice

These examples cover a range of complexities in dealing with equations of circles. To further solidify your understanding, try the following:

• Practice more problems: Seek additional practice problems in your textbook or online resources. Focus on problems that challenge you, especially those involving completing the square.
• Graphing circles: Graph the circles you solve to visualize their properties and confirm your calculations.
• Explore variations: Search for problems involving tangents, intersecting circles, or circles within other shapes.

Mastering equations of circles is a fundamental building block in mathematics. By understanding the standard and general forms, completing the square technique, and practicing consistently, you'll build a strong foundation for more advanced geometric concepts. Remember to always check your work and visualize the results—it will enhance your understanding and problem-solving abilities significantly.