Which equation represents a circle with a center at $(-4,9)$ and a diameter of 10 units?
A. $(x-9)^2+(y+4)^2=25$
B. $(x+4)^2+(y-9)^2=25$
C. $(x-9)^2+(y+4)^2=100$
D. $(x+4)^2+(y-9)^2=100
Answer (1)
Recall the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitute the given center ( − 4 , 9 ) into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = r 2 .
Calculate the radius from the diameter: r = 2 10 = 5 .
Substitute the radius into the equation to get the final answer: ( x + 4 ) 2 + ( y − 9 ) 2 = 25 .
Explanation
Problem Analysis The problem provides the center and diameter of a circle and asks for the correct equation representing that circle. The center is given as ( − 4 , 9 ) and the diameter is 10 units.
Recall Circle Equation The general equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Substitute Center Coordinates In our case, the center is ( − 4 , 9 ) , so h = − 4 and k = 9 . Substituting these values into the general equation, we get: ( x − ( − 4 ) ) 2 + ( y − 9 ) 2 = r 2 ( x + 4 ) 2 + ( y − 9 ) 2 = r 2
Calculate Radius The diameter is given as 10 units. The radius is half of the diameter, so: r = 2 10 = 5
Substitute Radius Value Now, substitute the radius r = 5 into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = 5 2 ( x + 4 ) 2 + ( y − 9 ) 2 = 25
Identify Correct Equation Comparing this equation with the given options, we find that the correct equation is: ( x + 4 ) 2 + ( y − 9 ) 2 = 25
Final Answer Therefore, the equation representing a circle with a center at ( − 4 , 9 ) and a diameter of 10 units is: ( x + 4 ) 2 + ( y − 9 ) 2 = 25
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the center and radius helps determine the placement and size of the garden. Similarly, in architecture, circular arches and domes rely on the principles of circle equations to ensure structural integrity and aesthetic appeal. In navigation, the equation of a circle can be used to define the range of a radar system or the area covered by a radio tower.
