Finding The Equation Of A Circle Passing Through Two Points And With Center On A Line
In the realm of coordinate geometry, circles hold a special place. Their elegant symmetry and fundamental properties make them a frequent subject of mathematical inquiry. Question 11 presents a fascinating challenge involving a circle defined by specific conditions. To truly grasp and conquer this problem, we will embark on a detailed journey, dissecting each element and employing powerful techniques to arrive at the correct solution. This guide is designed not just to provide the answer, but to empower you with a deep understanding of the underlying concepts, ensuring you can confidently tackle similar problems in the future.
Understanding the Problem
The question presents us with a circle that gracefully navigates through two distinct points, A(1, 2) and B(3, 4). These points act as anchors, partially defining the circle's position and size. However, the information doesn't stop there. We are given a crucial clue about the circle's center: it resides on the line y = 2x - 1. This linear equation acts as a constraint, limiting the possible locations of the circle's center. Our mission is to unveil the circle's equation, revealing its precise position and dimensions in the coordinate plane. The equation will be in the standard form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. Unraveling these values will unlock the circle's complete description.
To effectively solve this problem, we will employ a combination of geometric principles and algebraic techniques. The key lies in leveraging the given information strategically. First, we'll utilize the fact that the center of the circle lies on the line y = 2x - 1 to express the center's coordinates in terms of a single variable. This simplification will streamline our calculations. Next, we'll harness the property that the distance from the center to any point on the circle is equal to the radius. By applying this principle to points A and B, we'll establish equations that relate the center's coordinates and the radius. Solving this system of equations will reveal the precise location of the center and the value of the radius, ultimately leading us to the circle's equation. This step-by-step approach will not only yield the answer but also illuminate the elegant interplay between geometric concepts and algebraic manipulations.
Developing the Strategy
Our strategy hinges on a powerful combination of geometric insight and algebraic manipulation. To begin, let's represent the center of the circle as (h, k). Since this center lies on the line y = 2x - 1, we can express the y-coordinate, k, in terms of the x-coordinate, h, as k = 2h - 1. This elegant substitution reduces the unknowns, simplifying our calculations significantly. Now, the center is effectively defined by a single variable, h, making the problem more tractable.
The next crucial step involves leveraging the fundamental property of a circle: the distance from the center to any point on the circumference is constant and equal to the radius. Since points A(1, 2) and B(3, 4) lie on the circle, their distances from the center (h, 2h - 1) must be equal. We can express these distances using the distance formula, a direct application of the Pythagorean theorem. This formula, √[(x₂ - x₁)² + (y₂ - y₁)²], provides a precise measure of the distance between two points in the coordinate plane. By equating the distances from the center to A and the center to B, we establish an equation that relates the unknown coordinate h to itself. This equation forms the cornerstone of our solution.
After solving for the value of 'h', we will then substitute it back into our expression for the center's coordinates (h, 2h - 1) to determine the precise location of the circle's center. With the center's coordinates in hand, we can calculate the radius by computing the distance from the center to either point A or point B. Once we have both the center (h, k) and the radius r, we can confidently construct the equation of the circle in its standard form: (x - h)² + (y - k)² = r². This equation provides a complete and concise description of the circle's position and size in the coordinate plane. This methodical approach, combining geometric principles with algebraic techniques, will guide us to the correct solution with clarity and precision.
Step-by-Step Solution
Now, let's embark on the step-by-step solution to unravel the mystery of this circle's equation.
1. Expressing the Center's Coordinates
As we established in our strategy, let the center of the circle be represented by the coordinates (h, k). The problem states that the center lies on the line y = 2x - 1. This crucial piece of information allows us to express k in terms of h. Substituting x = h and y = k into the equation of the line, we get k = 2h - 1. This elegant substitution reduces the number of unknowns, simplifying our subsequent calculations. The center of the circle can now be expressed as (h, 2h - 1), a significant step forward in our solution.
2. Equating Distances
Since points A(1, 2) and B(3, 4) lie on the circle, their distances from the center (h, 2h - 1) must be equal. This principle stems from the fundamental definition of a circle: all points on the circumference are equidistant from the center. We can use the distance formula to express these distances. The distance formula, √[(x₂ - x₁)² + (y₂ - y₁)²], is a direct application of the Pythagorean theorem and provides a precise measure of the distance between two points in the coordinate plane.
Let's calculate the distance from the center (h, 2h - 1) to point A(1, 2):
Distance to A = √[(h - 1)² + (2h - 1 - 2)²] = √[(h - 1)² + (2h - 3)²]
Similarly, let's calculate the distance from the center (h, 2h - 1) to point B(3, 4):
Distance to B = √[(h - 3)² + (2h - 1 - 4)²] = √[(h - 3)² + (2h - 5)²]
Since these distances are equal, we can set the expressions equal to each other:
√[(h - 1)² + (2h - 3)²] = √[(h - 3)² + (2h - 5)²]
This equation forms the cornerstone of our solution, relating the unknown coordinate h to itself.
3. Solving for h
To solve for h, we can square both sides of the equation to eliminate the square roots:
(h - 1)² + (2h - 3)² = (h - 3)² + (2h - 5)²
Now, let's expand the squared terms:
h² - 2h + 1 + 4h² - 12h + 9 = h² - 6h + 9 + 4h² - 20h + 25
Combining like terms, we get:
5h² - 14h + 10 = 5h² - 26h + 34
Now, we can simplify the equation by subtracting 5h² from both sides:
-14h + 10 = -26h + 34
Adding 26h to both sides gives:
12h + 10 = 34
Subtracting 10 from both sides yields:
12h = 24
Finally, dividing both sides by 12, we find:
h = 2
This crucial result reveals the x-coordinate of the circle's center.
4. Finding the Center and Radius
Now that we know h = 2, we can determine the complete coordinates of the center. Recalling that the center is (h, 2h - 1), we substitute h = 2:
Center = (2, 2(2) - 1) = (2, 3)
Thus, the center of the circle is located at (2, 3).
Next, we need to find the radius. The radius is the distance from the center to any point on the circle. Let's use point A(1, 2) for this calculation. Using the distance formula:
Radius = √[(2 - 1)² + (3 - 2)²] = √(1² + 1²) = √2
Therefore, the radius of the circle is √2.
5. Writing the Equation of the Circle
Finally, we have all the pieces to write the equation of the circle. The standard form of the equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Substituting our values, h = 2, k = 3, and r = √2, we get:
(x - 2)² + (y - 3)² = (√2)²
Simplifying, the equation of the circle is:
(x - 2)² + (y - 3)² = 2
This elegant equation completely describes the circle that passes through points A(1, 2) and B(3, 4) and has its center on the line y = 2x - 1.
Analyzing the Options
Now that we've derived the equation of the circle, let's compare it to the options provided:
- (x - 2)² + (y - 3)² = 2
- (x - 3)² + (y - 5)² = 10
- (x - 1)² + (y - 1)² = 8
Our derived equation, (x - 2)² + (y - 3)² = 2, perfectly matches option 1. Therefore, option 1 is the correct answer.
Key Takeaways
This problem beautifully illustrates the interplay between geometry and algebra. Here are some key takeaways from our journey:
- Utilizing Given Information: The most crucial step in solving any mathematical problem is to carefully analyze the given information and identify how it can be used to your advantage. In this case, the fact that the center lies on the line y = 2x - 1 was a game-changer, allowing us to express the center's coordinates in terms of a single variable.
- Distance Formula: The distance formula is a powerful tool for calculating distances in the coordinate plane. It's a direct application of the Pythagorean theorem and is essential for solving problems involving circles, lines, and other geometric figures.
- Standard Equation of a Circle: Understanding the standard form of the circle's equation, (x - h)² + (y - k)² = r², is crucial for representing and analyzing circles. Knowing this form allows you to quickly identify the center and radius of a circle, and vice versa.
- Combining Geometry and Algebra: Many mathematical problems require a combination of geometric intuition and algebraic manipulation. This problem is a prime example, where we used geometric properties of circles and distances to set up algebraic equations and solve for unknowns.
- Step-by-Step Approach: Breaking down a complex problem into smaller, manageable steps is often the key to success. Our step-by-step solution, from expressing the center's coordinates to writing the final equation, demonstrates the effectiveness of this approach.
By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of problems involving circles and coordinate geometry.
Conclusion
In conclusion, the equation of the circle that passes through points A(1, 2) and B(3, 4) and has its center on the line y = 2x - 1 is (x - 2)² + (y - 3)² = 2. This solution was achieved by strategically combining geometric principles and algebraic techniques. We expressed the center's coordinates in terms of a single variable, utilized the distance formula, and solved a system of equations. This problem serves as a valuable illustration of how mathematical concepts intertwine and how a systematic approach can lead to elegant solutions. By understanding the key takeaways from this problem, you'll be better prepared to navigate the fascinating world of coordinate geometry and conquer future challenges.
