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)
Identify the center ( − 4 , 9 ) and calculate the radius r = 5 from the diameter.
Recall the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the center and radius into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = 25 .
The equation representing the circle is ( x + 4 ) 2 + ( y − 9 ) 2 = 25 .
Explanation
Identify the center and radius The center of the circle is at ( − 4 , 9 ) . The diameter of the circle is 10 units, so the radius is half of the diameter, which is 5 units.
Recall the standard equation of a circle The standard equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Substitute the center coordinates Substitute the given center ( − 4 , 9 ) for ( h , k ) into the equation: ( x − ( − 4 ) ) 2 + ( y − 9 ) 2 = r 2 ( x + 4 ) 2 + ( y − 9 ) 2 = r 2
Calculate the radius Calculate the radius r from the given diameter of 10 units: r = 2 10 = 5
Substitute the radius into the equation Substitute the radius r = 5 into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = 5 2 ( x + 4 ) 2 + ( y − 9 ) 2 = 25
State the final equation The equation of the circle with center ( − 4 , 9 ) and radius 5 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. This knowledge also extends to fields like computer graphics, where circles are fundamental elements in creating images and animations.
