The diameter of a circle has endpoints $(-3,-10)$ and $(4,-6)$. What is the equation of the circle?
$\left(x+\frac{1}{2}\right)^2+(y-8)^2=\frac{65}{4}$
$\left(x-\frac{1}{2}\right)^2+(y+8)^2=\frac{65}{4}$
$\left(x+\frac{1}{2}\right)^2+(y-8)^2=65$
$\left(x-\frac{1}{2}\right)^2+(y+8)^2=65
Answer (1)
Find the center of the circle by using the midpoint formula with the endpoints of the diameter: ( 2 − 3 + 4 , 2 − 10 + ( − 6 ) ) = ( 2 1 , − 8 ) .
Calculate the radius by finding the distance between the center and one of the endpoints using the distance formula: r = ( 4 − 2 1 ) 2 + ( − 6 − ( − 8 ) ) 2 = 4 65 .
Determine the equation of the circle using the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitute the center ( 2 1 , − 8 ) and r 2 = 4 65 into the equation, resulting in the final equation: ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65 .
Explanation
Problem Analysis The problem provides the endpoints of a circle's diameter and asks for the equation of the circle. To find the equation, we need to determine the center and the radius of the circle.
Finding the Center The center of the circle is the midpoint of the diameter. We can find the midpoint using the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Given the endpoints ( − 3 , − 10 ) and ( 4 , − 6 ) , we have: M = ( 2 − 3 + 4 , 2 − 10 + ( − 6 ) ) = ( 2 1 , 2 − 16 ) = ( 2 1 , − 8 ) So, the center of the circle is ( 2 1 , − 8 ) .
Finding the Radius The radius of the circle is the distance from the center to any point on the circle. We can use the distance formula to find the radius. Let's use the endpoint ( 4 , − 6 ) and the center ( 2 1 , − 8 ) :
r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 r = ( 4 − 2 1 ) 2 + ( − 6 − ( − 8 ) ) 2 r = ( 2 8 − 2 1 ) 2 + ( 2 ) 2 r = ( 2 7 ) 2 + 4 r = 4 49 + 4 16 r = 4 65 So, the radius is 4 65 . Therefore, r 2 = 4 65 .
Writing the Equation of the Circle The equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2 In our case, the center is ( 2 1 , − 8 ) , so h = 2 1 and k = − 8 . The radius squared is r 2 = 4 65 . Plugging these values into the equation, we get: ( x − 2 1 ) 2 + ( y − ( − 8 ) ) 2 = 4 65 ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65 Thus, the equation of the circle is ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65 .
Examples
Circles are fundamental in many real-world applications, from designing gears and wheels in mechanical engineering to understanding orbits in astronomy. For instance, determining the equation of a circular path is crucial in GPS navigation systems, where satellites use circles to map locations on Earth. Also, in architecture, circular designs are often used for aesthetic and structural purposes, requiring precise calculations to ensure stability and visual appeal. Understanding the equation of a circle allows engineers and designers to create efficient and visually pleasing structures.
