A circle has its center at $(-2,5)$ and a radius of 4 units. What is the equation of the circle?
$(x+2)^2+(y-5)^2=16$
$(x+2)^2+(y+5)^2=16$
$(x+2)^2+(y-5)^2=4$
$(x-2)^2+(y+5)^2=4
Answer (1)
Recall the general equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( − 2 , 5 ) and radius 4 into the equation.
Simplify the equation to ( x + 2 ) 2 + ( y − 5 ) 2 = 4 2 .
Obtain the final equation of the circle: ( x + 2 ) 2 + ( y − 5 ) 2 = 16 .
Explanation
Recall the general equation of a 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 this problem, we are given the center of the circle as ( − 2 , 5 ) and the radius as 4 units.
Substitute the given values Substitute the given values into the equation: ( x − ( − 2 ) ) 2 + ( y − 5 ) 2 = 4 2
Simplify the equation Simplify the equation: ( x + 2 ) 2 + ( y − 5 ) 2 = 16
State the final answer The equation of the circle with center ( − 2 , 5 ) and radius 4 is: ( x + 2 ) 2 + ( y − 5 ) 2 = 16
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 allows you to determine the exact layout and boundaries. Similarly, in architecture, circular windows or domes require precise calculations using the circle's equation to ensure structural integrity and aesthetic appeal. This knowledge also extends to fields like astronomy, where understanding circular orbits is fundamental.
