Circle Equation Problem Solving Guide Find The Equation Of A Circle

In this article, we will explore how to solve a classic circle equation problem. We'll break down the concepts, apply the relevant formulas, and walk through the solution step by step. Our focus will be on understanding circle equations and their components, such as the center and radius. By the end of this guide, you'll be well-equipped to tackle similar problems with confidence. This includes understanding how to use the distance formula and the standard form of a circle equation to find the correct answer. So, let's dive in and unravel the intricacies of circle geometry!

The Problem: Finding the Equation of Circle C

Let's start with the problem at hand: Circle C has a center at (-2, 10) and contains the point P(10, 5). The question is, which equation represents circle C? To tackle this, we need to understand the standard equation of a circle and how to derive it from the given information. The standard form of a circle's equation is given by:

$$(x - h)^2 + (y - k)^2 = r^2$$

Where (h, k) represents the coordinates of the center of the circle, and r is the radius. In our case, the center of circle C is given as (-2, 10), which means h = -2 and k = 10. This information is crucial for plugging into the equation. We also have a point P(10, 5) that lies on the circle. This point will help us determine the radius of the circle. Remember, the radius is the distance from the center of the circle to any point on its circumference. Therefore, by finding the distance between the center (-2, 10) and the point P(10, 5), we can find the value of r. Once we have the radius and the center coordinates, we can easily form the equation of circle C. This problem effectively tests our understanding of the fundamental properties of circles and their algebraic representation. Now, let's move on to the steps needed to find the radius and, subsequently, the correct equation.

Step 1: Calculate the Radius Using the Distance Formula

To find the equation of the circle, the first crucial step is to determine the radius. We know the center of the circle is at (-2, 10) and a point P(10, 5) lies on the circle. The radius, r, is the distance between these two points. We can calculate this distance using the distance formula, which is derived from the Pythagorean theorem.

The distance formula is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. In our case, (x₁, y₁) is the center (-2, 10) and (x₂, y₂) is the point P(10, 5). Plugging these values into the formula, we get:

$$r = \sqrt{(10 - (-2))^2 + (5 - 10)^2}$$

Simplifying the expression inside the square root:

$$r = \sqrt{(10 + 2)^2 + (5 - 10)^2}$$

$$r = \sqrt{(12)^2 + (-5)^2}$$

$$r = \sqrt{144 + 25}$$

$$r = \sqrt{169}$$

Therefore, the radius, r, is:

$$r = 13$$

Now that we have found the radius, which is 13 units, we can proceed to the next step: finding r², which we will need for the equation of the circle. Calculating r² is straightforward. We simply square the value of r:

$$r^2 = 13^2$$

$$r^2 = 169$$

This value, 169, will be a key component in determining the correct equation for circle C. Understanding the application of the distance formula is fundamental in solving geometry problems involving circles and other shapes. In this case, it allowed us to bridge the gap between the geometric definition of a radius and its algebraic value. With the radius and the center coordinates in hand, we are now ready to construct the equation of the circle.

Step 2: Construct the Equation of the Circle

With the radius r = 13 and the center (-2, 10) now determined, we have all the necessary components to construct the equation of circle C. Recall the standard form of the equation of a circle:

$$(x - h)^2 + (y - k)^2 = r^2$$

Where (h, k) is the center of the circle and r is the radius. We know that h = -2, k = 10, and r² = 169. Plugging these values into the standard equation, we get:

$$(x - (-2))^2 + (y - 10)^2 = 169$$

Simplifying the equation, we have:

$$(x + 2)^2 + (y - 10)^2 = 169$$

This is the equation of circle C. This equation represents all the points (x, y) that lie on the circle with a center at (-2, 10) and a radius of 13. It's important to understand that this equation is a direct algebraic representation of the geometric properties of the circle. Each term in the equation corresponds to a specific aspect of the circle: the (x + 2)² term relates to the horizontal distance from the center, the (y - 10)² term relates to the vertical distance from the center, and the 169 represents the square of the radius, which defines the size of the circle. Constructing the equation of a circle is a fundamental skill in analytic geometry. It allows us to describe geometric shapes using algebraic expressions, enabling us to analyze and manipulate them using algebraic techniques. In our case, we started with geometric information – the center and a point on the circle – and used it to derive the equation, thus establishing a concrete link between geometry and algebra. Now that we have the equation, let's compare it with the given options to identify the correct answer.

Step 3: Identify the Correct Option

Having derived the equation of circle C as:

$$(x + 2)^2 + (y - 10)^2 = 169$$

We now need to compare this equation with the given options to identify the correct one. The options provided are:

A. $(x - 2)^2 + (y + 10)^2 = 13$ B. $(x - 2)^2 + (y + 10)^2 = 169$ C. $(x + 2)^2 + (y - 10)^2 = 13$ D. $(x + 2)^2 + (y - 10)^2 = 169$

By direct comparison, we can see that our derived equation matches option D:

$$(x + 2)^2 + (y - 10)^2 = 169$$

Therefore, option D is the correct answer. Let's quickly analyze why the other options are incorrect:

• Option A: $(x - 2)^2 + (y + 10)^2 = 13$ has the wrong signs for the center coordinates (h and k) and the wrong value for r². It suggests a center at (2, -10) and a radius of √13, neither of which matches the given information.
• Option B: $(x - 2)^2 + (y + 10)^2 = 169$ has the correct value for r² but the wrong signs for the center coordinates. It represents a circle with a center at (2, -10) and a radius of 13, which is incorrect.
• Option C: $(x + 2)^2 + (y - 10)^2 = 13$ has the correct signs for the center coordinates but the wrong value for r². It represents a circle with a center at (-2, 10) and a radius of √13, which is also incorrect.

Only option D accurately represents a circle with a center at (-2, 10) and a radius of 13. This step of comparing the derived equation with the given options is crucial in problem-solving. It ensures that we have not made any errors in our calculations and that we have correctly applied the concepts and formulas. With the correct option identified, we can confidently conclude that we have successfully solved the problem.

Conclusion: Mastering Circle Equations

In this detailed walkthrough, we successfully solved the problem of finding the equation of a circle given its center and a point on its circumference. We began by understanding the standard form of a circle equation, which is:

$$(x - h)^2 + (y - k)^2 = r^2$$

Where (h, k) represents the center and r is the radius. We then used the distance formula to calculate the radius, which is the distance between the center (-2, 10) and the point P(10, 5). This gave us a radius of 13 units. Squaring the radius, we found r² = 169, which is a crucial component of the circle's equation. Next, we plugged the values of the center and r² into the standard equation to construct the equation of circle C:

$$(x + 2)^2 + (y - 10)^2 = 169$$

Finally, we compared our derived equation with the given options and correctly identified option D as the answer. This problem exemplifies the importance of understanding the relationship between the geometric properties of a circle and its algebraic representation. By mastering the concepts of the center, radius, and the distance formula, we can confidently solve a wide range of problems involving circle equations. Moreover, this process highlights the power of analytical geometry in bridging the gap between geometry and algebra. Understanding these concepts and practicing similar problems will enhance your problem-solving skills in mathematics and prepare you for more advanced topics in geometry and beyond. Remember, the key to success in mathematics is a solid understanding of the fundamentals and consistent practice. Keep exploring, keep learning, and keep solving!

Find the equation of a circle with center (-2, 10) that passes through the point (10, 5).

Circle Equation Problem Solving Guide: Find the Equation of a Circle