Give the equation of a circle with a diameter that has endpoints (-4, -4) and (-9, 7).
Answer (1)
Find the center of the circle using the midpoint formula: ( 2 x 1 + x 2 , 2 y 1 + y 2 ) . The center is ( − 6.5 , 1.5 ) .
Calculate the radius using the distance formula between the center and one endpoint: ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . The radius is 36.5 .
Write the equation of the circle in standard form: ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitute the center and radius into the standard equation to get the final answer: ( x + 6.5 ) 2 + ( y − 1.5 ) 2 = 36.5 .
Explanation
Problem Analysis The problem asks for the equation of a circle given the endpoints of a diameter. To find this, we need to determine the center and the radius of the circle. The center is the midpoint of the diameter, and the radius is the distance from the center to either endpoint of the diameter.
Finding the Center First, let's find the center of the circle. The midpoint formula is given by: ( 2 x 1 + x 2 , 2 y 1 + y 2 ) where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the endpoints of the diameter. In our case, ( x 1 , y 1 ) = ( − 4 , − 4 ) and ( x 2 , y 2 ) = ( − 9 , 7 ) . Plugging these values into the midpoint formula, we get: ( 2 − 4 + ( − 9 ) , 2 − 4 + 7 ) = ( 2 − 13 , 2 3 ) = ( − 6.5 , 1.5 ) So, the center of the circle is ( − 6.5 , 1.5 ) .
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 ( − 6.5 , 1.5 ) and one of the endpoints, say ( − 4 , − 4 ) . The distance formula is: ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Plugging in the values, we get: ( − 4 − ( − 6.5 ) ) 2 + ( − 4 − 1.5 ) 2 = ( 2.5 ) 2 + ( − 5.5 ) 2 = 6.25 + 30.25 = 36.5 ≈ 6.04 So, the radius of the circle is 36.5 .
Writing the Equation of the Circle Now we can write the equation of the circle in the standard form: ( x − h ) 2 + ( y − k ) 2 = r 2 where ( h , k ) is the center and r is the radius. In our case, ( h , k ) = ( − 6.5 , 1.5 ) and r = 36.5 . Plugging these values into the equation, we get: ( x − ( − 6.5 ) ) 2 + ( y − 1.5 ) 2 = ( 36.5 ) 2 ( x + 6.5 ) 2 + ( y − 1.5 ) 2 = 36.5 Thus, the equation of the circle is ( x + 6.5 ) 2 + ( y − 1.5 ) 2 = 36.5 .
Examples
Circles are fundamental in many real-world applications. For instance, in architecture, arches and domes often utilize circular geometry for structural integrity and aesthetic appeal. In navigation, understanding circles is crucial for determining distances and bearings on maps, as well as for calculating the range of radar or sonar systems. Even in art and design, circles are used to create balanced and harmonious compositions, demonstrating their versatility and importance across various fields.
