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 (2)
Find the center of the circle using the midpoint formula: ( 2 x 1 + x 2 , 2 y 1 + y 2 ) . The center is ( 2 1 , − 8 ) .
Calculate the radius using the distance formula between the center and one endpoint: r = ( 4 − 2 1 ) 2 + ( − 6 − ( − 8 ) ) 2 = 4 65 .
Determine r 2 : r 2 = 4 65 .
Write the equation of the circle using the center and radius: ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65 .
Explanation
Problem Analysis The diameter of the circle has endpoints ( − 3 , − 10 ) and ( 4 , − 6 ) . We need to find the equation of the circle.
Circle Equation The 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.
Finding the Center First, we need to find the center of the circle, which is the midpoint of the diameter. The midpoint formula is given by ( 2 x 1 + x 2 , 2 y 1 + y 2 ) . Using the given endpoints ( − 3 , − 10 ) and ( 4 , − 6 ) , we have:
h = 2 − 3 + 4 = 2 1 k = 2 − 10 + ( − 6 ) = 2 − 16 = − 8
So, the center of the circle is ( 2 1 , − 8 ) .
Finding the Radius Next, we need to find the radius of the circle. The radius is the distance from the center to any point on the circle. We can use the distance formula to find the distance between the center ( 2 1 , − 8 ) and one of the endpoints, say ( 4 , − 6 ) . The distance formula is given by ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
r = ( 4 − 2 1 ) 2 + ( − 6 − ( − 8 ) ) 2 r = ( 2 8 − 2 1 ) 2 + ( − 6 + 8 ) 2 r = ( 2 7 ) 2 + ( 2 ) 2 r = 4 49 + 4 r = 4 49 + 4 16 r = 4 65
So, the radius is 4 65 .
Finding r^2 Now, we need to find r 2 :
r 2 = ( 4 65 ) 2 = 4 65
Equation of the Circle Finally, we can write the equation of the circle using the center ( 2 1 , − 8 ) and r 2 = 4 65 :
( x − 2 1 ) 2 + ( y − ( − 8 ) ) 2 = 4 65 ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65
Therefore, the equation of the circle is ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65
Examples
Understanding the equation of a circle is very useful in many real-world applications. For example, in GPS navigation, your location is determined by calculating the intersection of circles from multiple satellites. Each satellite transmits a signal that indicates your distance from it, which defines a circle (or sphere in 3D). The GPS receiver then solves for the point where these circles intersect, pinpointing your exact location on Earth. This principle relies heavily on the circle equation and coordinate geometry.
The equation of the circle with a diameter defined by the endpoints ( − 3 , − 10 ) and ( 4 , − 6 ) is ( x − 2 1 ) 2 + ( y + 8 ) 2 = 4 65 . This is derived by calculating the center and radius from the endpoints. The answer corresponds to the equation provided in the second choice.
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