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Determination of the Center and Equation of a Circle Given Its Diameter End-Points
Determination of the Center and Equation of a Circle Given Its Diameter End-Points
This article aims to explain how to determine the center and equation of a circle when its diameter's end-points are provided. The process involves various geometric calculations that are fundamental in understanding the properties of a circle. This topic is particularly relevant for students and professionals in mathematics and related fields, as well as for those involved in data analysis and modeling.
The Problem
The question posed is: If A(5, 7) and B(9, 3) are the end-points of a diameter of a circle, what is the center of this circle?
Step 1: Finding the Center of the Circle
The midpoint of the diameter is the center of the circle. The formula for the midpoint, given two points (x1, y1) and (x2, y2), is:
Midpoint ( left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right) )
Substituting the points A(5, 7) and B(9, 3) into the formula, we get:
( x_c frac{5 9}{2} frac{14}{2} 7 ) ( y_c frac{7 3}{2} frac{10}{2} 5 )Therefore, the center of the circle is ( (7, 5) ).
Step 2: Deriving the Equation of the Circle
The general equation of a circle is given by:
( (x - h)^2 (y - k)^2 r^2 )
Where (h, k) is the center of the circle and r is the radius. From the previous step, we know the center (h, k) (7, 5).
Determining the Radius
The length of the radius can be found using the distance formula between the center and any point on the circle. We can use the center (7, 5) and one of the end-points of the diameter, say A(5, 7).
The distance formula between two points (x1, y1) and (x2, y2) is:
( d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} )
Substituting the points (7, 5) and (5, 7) into the formula, we get:
( d sqrt{(7 - 5)^2 (5 - 7)^2} ) ( d sqrt{2^2 (-2)^2} ) ( d sqrt{4 4} ) ( d sqrt{8} ) ( d 2sqrt{2} )Therefore, the radius of the circle is ( 2sqrt{2} ) units.
Final Equation of the Circle
Substituting the center (7, 5) and radius ( 2sqrt{2} ) into the general equation of the circle, we get:
( (x - 7)^2 (y - 5)^2 (2sqrt{2})^2 )
( (x - 7)^2 (y - 5)^2 8 )
Thus, the equation of the circle is: ( (x - 7)^2 (y - 5)^2 8 ).
Summary and Conclusion
In summary, we have determined that the center of the circle with end-points A(5, 7) and B(9, 3) is (7, 5). We also derived the equation of the circle: ( (x - 7)^2 (y - 5)^2 8 ).
This problem illustrates the application of geometric principles and algebraic techniques to solve real-world problems. Understanding these concepts is crucial for students and professionals in mathematics and related fields.