Equation Of A Circle Passing Through Two Points A(1 2) And B(3 4)
In this article, we will delve into the process of finding the equation of a circle that satisfies specific conditions. Our focus will be on a circle that passes through two given points, A(1, 2) and B(3, 4), with the added constraint that its center lies on the line y = 2x - 1. This problem combines concepts from coordinate geometry, including the distance formula, the equation of a circle, and linear equations. Understanding the interplay between these concepts is crucial for solving this type of problem. We will break down the solution step-by-step, providing a clear and concise explanation for each step. By the end of this article, you will have a solid understanding of how to approach similar problems involving circles and their equations.
Understanding the Problem
Before diving into the solution, let's clearly understand the problem statement. We are given two points, A(1, 2) and B(3, 4), and a line, y = 2x - 1. Our goal is to find the equation of a circle that passes through both points A and B, and whose center lies on the given line. The general equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². Therefore, to determine the equation of the circle, we need to find the coordinates of the center (h, k) and the radius r. The condition that the center lies on the line y = 2x - 1 provides a crucial relationship between h and k, which we will use to solve the problem. Additionally, the fact that the circle passes through points A and B implies that the distance from the center to each of these points is equal to the radius of the circle. This gives us two more equations involving h, k, and r, which we can use to solve for the unknowns. The problem requires a combination of algebraic manipulation and geometric understanding.
Step-by-Step Solution
H2: Step 1: Define the Center and Use the Line Equation
Let the center of the circle be (h, k). Since the center lies on the line y = 2x - 1, we can write the relationship between h and k as: k = 2h - 1. This equation is our first key to solving the problem. It expresses one coordinate of the center in terms of the other, effectively reducing the number of unknowns. This step is crucial because it incorporates the given condition that the center lies on a specific line. Without this condition, the problem would have infinitely many solutions. By using this relationship, we can express the center of the circle in terms of a single variable, h. This simplifies the subsequent calculations and allows us to solve for the unknowns more easily. The equation k = 2h - 1 is a linear equation, and it represents a straight line in the coordinate plane. The center of the circle must lie on this line, which means that the coordinates of the center must satisfy this equation. This constraint is essential for finding the unique circle that satisfies the given conditions. The next steps will involve using the distance formula and the fact that the distances from the center to points A and B are equal to the radius of the circle. This will give us two more equations involving h and k, which we can then solve simultaneously with the equation k = 2h - 1 to find the coordinates of the center.
H2: Step 2: Apply the Distance Formula
Since the circle passes through points A(1, 2) and B(3, 4), the distance from the center (h, k) to each of these points is equal to the radius, r. We can use the distance formula to express these distances:
- Distance from (h, k) to A(1, 2): √((h - 1)² + (k - 2)²) = r
- Distance from (h, k) to B(3, 4): √((h - 3)² + (k - 4)²) = r
Squaring both sides of each equation to eliminate the square root, we get:
- (h - 1)² + (k - 2)² = r²
- (h - 3)² + (k - 4)² = r²
Now we have two equations where both left-hand sides are equal to r². This implies that they must be equal to each other. This is a crucial step in the solution because it eliminates the variable r, leaving us with an equation involving only h and k. By equating the two expressions, we can find a relationship between h and k that is independent of the radius. This relationship, combined with the equation k = 2h - 1 from Step 1, will allow us to solve for the values of h and k. The distance formula is a fundamental tool in coordinate geometry, and it is used extensively in problems involving distances between points. In this case, it allows us to relate the coordinates of the center of the circle to the coordinates of the points that lie on the circle. The squaring of both sides of the equations is a common technique used to simplify equations involving square roots. It eliminates the square root and makes the equations easier to work with.
H2: Step 3: Equate the Distances and Simplify
Equating the two expressions for r² obtained in Step 2, we have:
(h - 1)² + (k - 2)² = (h - 3)² + (k - 4)²
Expanding the squares, we get:
h² - 2h + 1 + k² - 4k + 4 = h² - 6h + 9 + k² - 8k + 16
Now, we can simplify the equation by canceling out the h² and k² terms and rearranging the remaining terms:
-2h + 1 - 4k + 4 = -6h + 9 - 8k + 16
4h + 4k = 20
Dividing both sides by 4, we get:
h + k = 5
This simplified equation provides another relationship between h and k. This step is essential because it leads to a linear equation in h and k, which can be easily solved in conjunction with the equation k = 2h - 1 from Step 1. By equating the distances, we have effectively used the geometric condition that the distances from the center to the two points on the circle must be equal. This condition is a direct consequence of the definition of a circle, which is the set of all points equidistant from the center. The simplification process involves algebraic manipulation, which requires careful attention to detail to avoid errors. The expansion of the squares and the rearrangement of terms are standard algebraic techniques that are used frequently in problem-solving. The resulting equation, h + k = 5, is a linear equation, which represents a straight line in the coordinate plane. This line is the perpendicular bisector of the line segment joining the points A and B. The center of the circle must lie on this line, which is another way of understanding the geometric significance of this equation.
H2: Step 4: Solve the System of Equations
We now have a system of two linear equations in two variables, h and k:
- k = 2h - 1
- h + k = 5
We can solve this system using substitution or elimination. Let's use substitution. Substitute the expression for k from equation (1) into equation (2):
h + (2h - 1) = 5
3h - 1 = 5
3h = 6
h = 2
Now, substitute h = 2 back into equation (1) to find k:
k = 2(2) - 1
k = 4 - 1
k = 3
Thus, the center of the circle is (2, 3). Solving the system of equations is a crucial step in finding the coordinates of the center of the circle. The system of equations represents the intersection of two lines in the coordinate plane. The solution to the system is the point where the two lines intersect, which is the center of the circle in this case. The substitution method is a common technique for solving systems of equations. It involves expressing one variable in terms of the other and then substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can be easily solved. The values of h and k that we have found, h = 2 and k = 3, represent the x-coordinate and y-coordinate of the center of the circle, respectively. These values are essential for determining the equation of the circle. The next step will involve finding the radius of the circle using the distance formula.
H2: Step 5: Calculate the Radius
Now that we have the center (h, k) = (2, 3), we can calculate the radius, r, using the distance formula with either point A or point B. Let's use point A(1, 2):
r = √((2 - 1)² + (3 - 2)²)
r = √(1² + 1²)
r = √2
Therefore, r² = 2. Calculating the radius is the final step in determining the equation of the circle. The radius is the distance from the center of the circle to any point on the circle. We can use either point A or point B to calculate the radius since both points lie on the circle. The distance formula is used to calculate the distance between two points in the coordinate plane. In this case, we are using it to calculate the distance between the center of the circle and a point on the circle. The square of the radius, r², is also important because it appears in the equation of the circle. We have calculated r = √2, so r² = 2. This value will be used in the final step to write the equation of the circle.
H2: Step 6: Write the Equation of the Circle
Using the center (2, 3) and r² = 2, the equation of the circle is:
(x - 2)² + (y - 3)² = 2
This matches option 1. Writing the equation of the circle is the final step in solving the problem. The equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². We have found the coordinates of the center, (h, k) = (2, 3), and the square of the radius, r² = 2. Substituting these values into the general equation of a circle gives us the equation of the specific circle that satisfies the given conditions. The equation (x - 2)² + (y - 3)² = 2 represents a circle with center (2, 3) and radius √2. This is the equation of the circle that passes through the points A(1, 2) and B(3, 4) and whose center lies on the line y = 2x - 1. This completes the solution to the problem.
Conclusion
In conclusion, we have successfully found the equation of the circle that passes through points A(1, 2) and B(3, 4) and whose center lies on the line y = 2x - 1. The equation of the circle is (x - 2)² + (y - 3)² = 2. This problem demonstrates the interplay between coordinate geometry concepts, including the distance formula, the equation of a circle, and linear equations. By breaking down the problem into smaller steps and carefully applying the relevant formulas and techniques, we were able to arrive at the solution. This type of problem is common in mathematics and requires a solid understanding of the fundamental concepts of coordinate geometry. The key to solving this problem was to use the given information to set up a system of equations and then solve that system to find the unknowns. The steps involved in the solution included: defining the center of the circle, using the line equation to relate the coordinates of the center, applying the distance formula to relate the distances from the center to the points on the circle, equating the distances to eliminate the radius, simplifying the resulting equation, solving the system of equations to find the coordinates of the center, calculating the radius, and finally, writing the equation of the circle. By mastering these techniques, you will be well-equipped to solve similar problems in the future.
