Circle Equation Explained Step-by-Step Solution
In the realm of analytical geometry, circles stand as fundamental geometric figures, their equations elegantly capturing their symmetrical nature. This comprehensive guide delves into the intricacies of determining the equation of a circle, specifically focusing on the scenario where the center and a point on the circle are known. We will meticulously dissect the problem, providing a step-by-step solution while elucidating the underlying concepts and principles. By the end of this exploration, you will not only be equipped to solve this particular problem but also possess a profound understanding of the equation of a circle and its applications. Let's embark on this mathematical journey together!
Decoding the Circle's Essence
Before we embark on the solution, let's solidify our understanding of the fundamental equation of a circle. A circle, in its essence, is defined as the locus of all points equidistant from a fixed point, known as the center. This constant distance is termed the radius, a crucial parameter in defining the circle's dimensions. The equation of a circle, derived from the Pythagorean theorem, elegantly captures this geometric relationship. In a Cartesian coordinate system, the equation of a circle with center (h, k) and radius r is expressed as:
(x - h)² + (y - k)² = r²
This equation forms the cornerstone of our analysis, providing a direct link between the circle's geometric properties and its algebraic representation. The variables x and y represent the coordinates of any point lying on the circle's circumference, while h and k denote the x and y coordinates of the circle's center, respectively. The radius, r, quantifies the distance from the center to any point on the circle. Understanding this equation is paramount to solving problems related to circles, including the one at hand.
Step 1 Identifying the Center and a Point
The problem statement provides us with two key pieces of information: the center of circle C, which is located at the coordinates (-2, 10), and a point P(10, 5) that lies on the circle's circumference. These two data points are the foundation upon which we will construct the circle's equation. The center, (-2, 10), directly provides us with the values of h and k, which are -2 and 10, respectively. The point P(10, 5) will be instrumental in determining the circle's radius, r. By leveraging the distance formula, we can calculate the distance between the center and point P, thereby obtaining the radius. This initial step sets the stage for the subsequent calculations, allowing us to systematically determine the equation of the circle.
Step 2 Calculating the Radius
The radius of a circle, as we've established, is the distance between the center and any point on its circumference. In this case, we have the center (-2, 10) and a point P(10, 5) on the circle. To calculate the distance between these two points, we employ the distance formula, a direct application of the Pythagorean theorem:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Here, (x₁, y₁) represents the coordinates of the center, (-2, 10), and (x₂, y₂) represents the coordinates of point P, (10, 5). Substituting these values into the distance formula, we get:
Distance = √[(10 - (-2))² + (5 - 10)²]
Simplifying the expression within the square root:
Distance = √[(12)² + (-5)²]
Distance = √(144 + 25)
Distance = √169
Distance = 13
Thus, the radius of circle C is determined to be 13 units. This value is crucial for completing the circle's equation, as it represents the constant distance between the center and any point on the circle.
Step 3 Constructing the Equation
Now that we have the center (-2, 10) and the radius 13, we possess all the necessary components to construct the equation of circle C. Recall the general equation of a circle:
(x - h)² + (y - k)² = r²
Substituting the values we've obtained, where h = -2, k = 10, and r = 13, into the equation, we get:
(x - (-2))² + (y - 10)² = 13²
Simplifying the equation:
(x + 2)² + (y - 10)² = 169
This equation represents circle C, encapsulating its geometric properties in a concise algebraic form. The equation clearly indicates the circle's center at (-2, 10) and its radius of 13 units. This equation is the ultimate answer to the problem, providing a complete description of the circle.
Dissecting the Answer Choices
Having derived the equation of circle C, let's now meticulously examine the provided answer choices to identify the correct one. The equation we obtained is:
(x + 2)² + (y - 10)² = 169
Now, let's compare this equation with the given options:
- A. (x - 2)² + (y + 10)² = 13 This equation has incorrect signs for the center's coordinates and an incorrect value for the radius squared.
- B. (x - 2)² + (y + 10)² = 169 This equation has incorrect signs for the center's coordinates but the correct value for the radius squared.
- C. (x + 2)² + (y - 10)² = 13 This equation has the correct signs for the center's coordinates but an incorrect value for the radius squared.
- D. (x + 2)² + (y - 10)² = 169 This equation perfectly matches the equation we derived, with the correct signs for the center's coordinates and the correct value for the radius squared.
Therefore, the correct answer is D. (x + 2)² + (y - 10)² = 169. This choice accurately represents the equation of circle C, with its center at (-2, 10) and a radius of 13 units. The process of comparing our derived equation with the answer choices reinforces our understanding and ensures the accuracy of our solution.
The Significance of the Circle's Equation
The equation of a circle, beyond being a mere algebraic expression, holds profound significance in various fields of mathematics and its applications. It serves as a fundamental tool in analytical geometry, allowing us to describe and analyze circles with precision. The equation's ability to capture the circle's center and radius in a concise form makes it invaluable for solving geometric problems, such as finding the intersection of circles, determining tangents, and calculating areas and circumferences. Moreover, the equation of a circle finds applications in diverse real-world scenarios. In physics, it is used to model circular motion, such as the orbits of planets around stars or the trajectory of a projectile. In engineering, it is employed in the design of circular structures, such as bridges and tunnels. In computer graphics, it is used to create and manipulate circular shapes. The versatility and applicability of the circle's equation underscore its importance in mathematics and its role in bridging the gap between abstract concepts and real-world phenomena.
Conclusion
In this comprehensive guide, we embarked on a journey to determine the equation of circle C, given its center and a point on its circumference. We meticulously dissected the problem, elucidating the underlying concepts and principles. We began by solidifying our understanding of the general equation of a circle, which forms the foundation of our analysis. We then identified the center and the point, calculated the radius using the distance formula, and constructed the equation of circle C. Finally, we compared our derived equation with the answer choices, confirming the correct answer as D. (x + 2)² + (y - 10)² = 169. Through this step-by-step solution, we not only solved the problem but also gained a deeper appreciation for the equation of a circle and its significance in mathematics and its applications. The ability to derive and interpret the equation of a circle is a valuable skill that empowers us to analyze and understand circular shapes in various contexts.
Select the Correct Answer for Circle Equation A Step-by-Step Solution
Let's tackle the question head-on, making sure we understand the underlying principles. The question presents us with a circle, cleverly named Circle C, nestled in the coordinate plane. We're given two crucial pieces of information: the circle's center, which resides at the coordinates (-2, 10), and a point, P(10, 5), that gracefully lies on the circle's circumference. Our mission, should we choose to accept it, is to decipher the circle's equation from the options provided. This isn't just about picking the right answer; it's about understanding the why behind it. We'll break down the process, ensuring that each step is clear and logical, making the journey as enlightening as the destination. So, let's dive in and unlock the secrets of Circle C!
Understanding the Circle Equation
At the heart of this problem lies the equation of a circle, a fundamental concept in coordinate geometry. This equation is a mathematical shorthand, a way to describe all the points that lie on the circle's edge. It's derived from the Pythagorean theorem, a cornerstone of geometry, and elegantly captures the relationship between a circle's center, its radius, and any point on its circumference. The general form of the equation is:
(x - h)² + (y - k)² = r²
Where:
- (h, k) represents the coordinates of the circle's center – the heart of the circle, if you will.
- r is the radius, the distance from the center to any point on the circle's edge.
- (x, y) are the coordinates of any point that lives on the circle itself.
Think of this equation as a blueprint for a circle. Knowing the center (h, k) and the radius (r) allows us to draw the circle perfectly. Conversely, if we have the equation, we can immediately identify the circle's center and radius. This equation is our key to unlocking the solution.
Step 1 Unpacking the Given Information
The question has already handed us some vital clues. We know the center of Circle C is at (-2, 10). This tells us that in our general equation, h = -2 and k = 10. It's like having two pieces of a puzzle already in place. The other key piece of information is point P(10, 5), which sits on the circle's rim. This point is our secret weapon for finding the radius. Remember, the radius is the distance from the center to any point on the circle. So, point P provides us with a specific distance to calculate, allowing us to find the 'r' in our equation. By carefully extracting this information, we've set ourselves up for the next step in our journey.
Step 2 The Radius Quest Using the Distance Formula
To find the radius, we'll employ the distance formula, a direct descendant of the Pythagorean theorem. The distance formula allows us to calculate the distance between two points in the coordinate plane. It's expressed as:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where:
- (x₁, y₁) are the coordinates of the first point (in our case, the circle's center).
- (x₂, y₂) are the coordinates of the second point (point P on the circle).
We'll plug in the coordinates of the center (-2, 10) and point P(10, 5) into this formula:
Distance = √[(10 - (-2))² + (5 - 10)²]
Let's simplify this step-by-step:
Distance = √[(12)² + (-5)²]
Distance = √(144 + 25)
Distance = √169
Distance = 13
Eureka! We've found the radius. The distance between the center and point P, and therefore the radius of Circle C, is 13 units. Now we have all the ingredients to bake our circle equation.
Step 3 Crafting the Circle Equation
With the center (h = -2, k = 10) and the radius (r = 13) in hand, we're ready to assemble the equation of Circle C. We'll revisit our general equation:
(x - h)² + (y - k)² = r²
And substitute our values:
(x - (-2))² + (y - 10)² = 13²
Let's simplify:
(x + 2)² + (y - 10)² = 169
This, my friends, is the equation of Circle C. It's a mathematical portrait of our circle, capturing its essence in a neat algebraic expression. Now, the final step is to compare this masterpiece with the answer choices and select the one that matches.
Deciphering the Answer Options
Now comes the moment of truth. We have our equation: (x + 2)² + (y - 10)² = 169. Let's carefully compare it to the options presented in the question. This is where attention to detail is key; a single misplaced sign or number can lead us astray.
- Option A: (x - 2)² + (y + 10)² = 13 - This is incorrect. The signs within the parentheses are wrong (should be +2 and -10), and the value on the right side is the radius, not the radius squared.
- Option B: (x - 2)² + (y + 10)² = 169 - This is also incorrect. While the 169 is correct (13 squared), the signs within the parentheses are still off.
- Option C: (x + 2)² + (y - 10)² = 13 - Close, but no cigar. The signs are correct, but the right side should be 169, not 13.
- Option D: (x + 2)² + (y - 10)² = 169 - Bingo! This perfectly matches the equation we derived. The signs are correct, and the right side is the correct radius squared.
Therefore, the correct answer is D. (x + 2)² + (y - 10)² = 169. We've successfully navigated the problem, understanding each step along the way.
Real-World Circles and Their Equations
The equation of a circle isn't just an abstract mathematical concept; it has real-world applications galore. Circles are fundamental shapes in the universe, from the orbits of planets to the wheels on our cars. Understanding their equations allows us to model and analyze these phenomena. In physics, the equation of a circle is used to describe circular motion, a key concept in understanding how things move in curved paths. In engineering, circles are used in the design of gears, wheels, and many other mechanical components. Architects use circles in building designs, creating aesthetically pleasing and structurally sound structures. Even in computer graphics, circles are essential for creating everything from simple icons to complex 3D models. The equation of a circle is a powerful tool that bridges the gap between abstract mathematics and the tangible world around us.
Concluding Thoughts
We've journeyed through the world of circles and their equations, unraveling the problem step-by-step. We've seen how the equation of a circle encapsulates its essential properties – its center and radius – in a concise algebraic form. We've used the distance formula to bridge the gap between geometry and algebra, calculating the radius from given points. And we've applied this knowledge to select the correct equation for Circle C. But more importantly, we've understood the why behind each step, solidifying our understanding of the concepts involved. The ability to manipulate and interpret the equation of a circle is a valuable skill, not just in mathematics, but in many fields that rely on geometric understanding. So, keep exploring, keep questioning, and keep unlocking the mathematical secrets of the world around you!
