Finding Circle Equation Passing Through Two Points With Center On A Line

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Introduction

In this article, we will delve into the solution of a fascinating geometry problem: finding the equation of a circle given certain conditions. Specifically, we are tasked with determining the equation of a circle that passes through two given points, A(1, 2) and B(3, 4), with its center lying on the line y = 2x - 1. This problem elegantly combines concepts from coordinate geometry, including the distance formula, the equation of a line, and the standard form of a circle's equation. Understanding these concepts is crucial for tackling various geometric problems, and this example provides an excellent opportunity to reinforce that knowledge. We will explore the underlying principles, derive the necessary equations, and methodically solve for the circle's center and radius. This detailed step-by-step approach will not only lead us to the correct answer but also enhance our problem-solving skills in coordinate geometry. By the end of this discussion, you will have a clear understanding of how to approach similar problems, combining geometric intuition with algebraic techniques to arrive at solutions. Let's embark on this geometrical journey and unravel the equation of the circle.

Problem Statement

The problem presents us with a geometrical challenge that requires careful application of coordinate geometry principles. We are given two points, A(1, 2) and B(3, 4), and the information that a circle passes through both of these points. Additionally, we know that the center of this circle lies on the line y = 2x - 1. Our primary objective is to determine the equation of this circle. This involves finding the coordinates of the center and the radius of the circle, which will allow us to express the equation in the standard form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is its radius. The challenge here lies in using the given information effectively to derive the values of h, k, and r. We need to utilize the fact that the distances from the center to points A and B are equal (since they are both radii of the same circle), and also incorporate the constraint that the center lies on the given line. This problem not only tests our understanding of circles and their equations but also our ability to combine different pieces of information to solve a geometrical puzzle. The solution will involve algebraic manipulations and a clear understanding of the relationships between points, lines, and circles in the coordinate plane. Let’s proceed to break down the problem and develop a step-by-step solution.

Solution Approach

To solve this problem effectively, we need a strategic approach that combines geometric principles with algebraic techniques. Our approach will be structured in a series of logical steps, ensuring that we utilize all the given information to its fullest potential. First, we will leverage the fact that the center of the circle lies on the line y = 2x - 1. This means that the coordinates of the center can be expressed in terms of a single variable. Let’s denote the center of the circle as (h, k). Since the center lies on the line, we can write k = 2h - 1. This simplifies our problem by reducing the number of unknowns. Next, we will use the distance formula, which is a cornerstone of coordinate geometry. The distance between the center of the circle and any point on the circle is equal to the radius. Since points A and B lie on the circle, the distances from the center (h, k) to A(1, 2) and B(3, 4) must be equal. We can set up an equation using the distance formula for these two distances. This equation will involve h, k, and the radius r. However, since we have already expressed k in terms of h, we can further simplify the equation. By equating the squared distances (to avoid dealing with square roots), we will obtain an algebraic equation in terms of h. Solving this equation will give us the value of h, and consequently, the value of k (using the relationship k = 2h - 1). Once we have the coordinates of the center (h, k), we can calculate the radius r by finding the distance from the center to either point A or point B. Finally, with the center and radius determined, we can write the equation of the circle in the standard form (x - h)^2 + (y - k)^2 = r^2. This step-by-step approach allows us to systematically unravel the problem, ensuring accuracy and clarity in our solution. Now, let’s delve into the detailed calculations and algebraic manipulations required to find the equation of the circle.

Step 1: Expressing the Center in Terms of a Single Variable

As outlined in our approach, the first crucial step is to express the coordinates of the circle's center in terms of a single variable. This simplification is made possible by the given condition that the center lies on the line y = 2x - 1. Let the center of the circle be denoted by the coordinates (h, k). Since the center lies on the line y = 2x - 1, we can directly substitute h for x and k for y in the equation of the line. This gives us the relationship k = 2h - 1. This equation is fundamental to our solution as it allows us to eliminate one variable, reducing the complexity of the problem. By expressing k in terms of h, we have effectively parameterized the center of the circle. Instead of dealing with two independent variables, h and k, we now have a single variable h that determines both coordinates of the center. This is a common and powerful technique in coordinate geometry, allowing us to translate geometric constraints into algebraic relationships. The equation k = 2h - 1 essentially encodes the information that the center must lie on the given line. This will be instrumental in subsequent steps when we use the distance formula to relate the center to the points A and B. By making this initial simplification, we set the stage for a more manageable algebraic solution. The ability to recognize and utilize such constraints is a key skill in solving geometry problems. With this relationship established, we can now proceed to the next phase of our solution, which involves using the distance formula to derive further equations.

Step 2: Applying the Distance Formula

With the center of the circle expressed in terms of a single variable, we now turn to the distance formula to establish further relationships. The core idea here is that the distance from the center of the circle to any point on the circle is equal to the radius. Since points A(1, 2) and B(3, 4) lie on the circle, their distances from the center (h, k) must be the same. The distance formula, which is derived from the Pythagorean theorem, states that the distance d between two points (x1, y1) and (x2, y2) is given by d = √((x2 - x1)^2 + (y2 - y1)^2). Applying this formula, we can express the distance between the center (h, k) and point A(1, 2) as √((h - 1)^2 + (k - 2)^2), and the distance between the center (h, k) and point B(3, 4) as √((h - 3)^2 + (k - 4)^2). Since these distances are equal (both being radii of the same circle), we can equate them: √((h - 1)^2 + (k - 2)^2) = √((h - 3)^2 + (k - 4)^2). To simplify this equation and avoid dealing with square roots, we can square both sides: (h - 1)^2 + (k - 2)^2 = (h - 3)^2 + (k - 4)^2. This equation now relates h and k without the complications of square roots. However, we recall that we have already established a relationship between h and k: k = 2h - 1. Substituting this expression for k into our distance equation will allow us to eliminate k and obtain an equation solely in terms of h. This substitution is a crucial step in simplifying the problem and moving closer to a solution. By applying the distance formula and utilizing the relationship between h and k, we are systematically reducing the unknowns and transforming the geometric problem into an algebraic one. The resulting equation in h will be the key to unlocking the coordinates of the center of the circle.

Step 3: Solving for the Center's Coordinates

Having established the equation (h - 1)^2 + (k - 2)^2 = (h - 3)^2 + (k - 4)^2, and knowing that k = 2h - 1, we can now proceed to solve for the center's coordinates. The first step is to substitute k = 2h - 1 into the equation derived from the distance formula. This gives us: (h - 1)^2 + (2h - 1 - 2)^2 = (h - 3)^2 + (2h - 1 - 4)^2. Simplifying this, we get: (h - 1)^2 + (2h - 3)^2 = (h - 3)^2 + (2h - 5)^2. Now, we need to expand the squared terms: (h^2 - 2h + 1) + (4h^2 - 12h + 9) = (h^2 - 6h + 9) + (4h^2 - 20h + 25). Combining like terms on each side, we have: 5h^2 - 14h + 10 = 5h^2 - 26h + 34. Notice that the 5h^2 terms cancel out, which simplifies the equation significantly. We are left with: -14h + 10 = -26h + 34. Now, we can isolate h by adding 26h to both sides and subtracting 10 from both sides: 12h = 24. Dividing both sides by 12, we find: h = 2. This is a crucial result, as it gives us the x-coordinate of the center of the circle. To find the y-coordinate k, we can use the relationship k = 2h - 1. Substituting h = 2, we get: k = 2(2) - 1 = 4 - 1 = 3. Therefore, the center of the circle is (2, 3). This is a significant milestone in our solution. We have successfully used the given information and algebraic manipulations to determine the coordinates of the circle's center. With the center known, we can now move on to the final step: finding the radius and writing the equation of the circle.

Step 4: Determining the Radius and the Circle's Equation

Now that we have found the center of the circle to be (2, 3), the final step is to determine the radius and then write the equation of the circle. The radius r is the distance from the center to any point on the circle. We can use either point A(1, 2) or point B(3, 4) to calculate the radius. Let's use point A(1, 2). The distance formula gives us: r = √((2 - 1)^2 + (3 - 2)^2) = √(1^2 + 1^2) = √2. So, the radius of the circle is √2. Now that we have both the center (h, k) = (2, 3) and the radius r = √2, we can write the equation of the circle in the standard form: (x - h)^2 + (y - k)^2 = r^2. Substituting the values we found, we get: (x - 2)^2 + (y - 3)^2 = (√2)^2. Simplifying, we have: (x - 2)^2 + (y - 3)^2 = 2. This is 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. We have successfully solved the problem by systematically using the given information and applying the principles of coordinate geometry. The equation (x - 2)^2 + (y - 3)^2 = 2 represents the circle we were looking for. This completes our solution.

Final Answer and Options

After a detailed step-by-step solution, we have determined that the equation of the circle is (x - 2)^2 + (y - 3)^2 = 2. Now, let's compare this result with the options provided in the problem statement.

Options:

  1. (x - 2)^2 + (y - 3)^2 = 2
  2. (x - 3)^2 + (y - 5)^2 = 10
  3. (x - 1)^2 + (y - 1)^2 = 8
  4. (x - 4)^2 + (y - 7)^2 = 26

By direct comparison, we can see that our derived equation matches option 1 perfectly. Therefore, the correct answer is option 1.

Conclusion

In conclusion, we have successfully found the equation of the circle that passes through the points A(1, 2) and B(3, 4) with its center lying on the line y = 2x - 1. The solution involved a careful application of coordinate geometry principles, including the distance formula and the equation of a line. We systematically worked through the problem, first expressing the center of the circle in terms of a single variable, then using the distance formula to relate the center to the given points, solving for the center's coordinates, and finally determining the radius and writing the equation of the circle. The correct equation is (x - 2)^2 + (y - 3)^2 = 2, which corresponds to option 1 in the provided choices. This problem serves as a good example of how geometric constraints can be translated into algebraic equations, and how these equations can be manipulated to find a solution. The ability to combine geometric intuition with algebraic techniques is crucial in solving a wide range of mathematical problems. We hope this detailed explanation has provided a clear understanding of the solution process and has enhanced your problem-solving skills in coordinate geometry.