Find an equation of the circle that satisfies the given conditions. Center $(4,-5); radius 4
Answer (1)
The problem provides the center ( 4 , − 5 ) and radius 4 of a circle.
We use 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.
Substituting the given values, we get ( x − 4 ) 2 + ( y + 5 ) 2 = 4 2 .
Simplifying, the equation of the circle is ( x − 4 ) 2 + ( y + 5 ) 2 = 16 .
Explanation
Problem Analysis We are given the center and radius of a circle and asked to find its equation. The center is ( 4 , − 5 ) and the radius is 4 . The general equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 . We will substitute the given values into this equation to find the equation of the circle.
Substitute Values Substitute the given values h = 4 , k = − 5 , and r = 4 into the general equation of a circle: ( x − 4 ) 2 + ( y − ( − 5 ) ) 2 = 4 2
Simplify Simplify the equation: ( x − 4 ) 2 + ( y + 5 ) 2 = 16
Expand the Equation The equation of the circle is ( x − 4 ) 2 + ( y + 5 ) 2 = 16 . We can also expand this equation to get it in the general form x 2 + y 2 + A x + B y + C = 0 . Expanding the equation, we get: ( x 2 − 8 x + 16 ) + ( y 2 + 10 y + 25 ) = 16 x 2 − 8 x + 16 + y 2 + 10 y + 25 − 16 = 0 x 2 + y 2 − 8 x + 10 y + 25 = 0
Final Answer The equation of the circle is ( x − 4 ) 2 + ( y + 5 ) 2 = 16 or x 2 + y 2 − 8 x + 10 y + 25 = 0 . The question asks for the equation of the circle, so either form is acceptable. The standard form is ( x − 4 ) 2 + ( y + 5 ) 2 = 16 .
Examples
Circles are fundamental in many real-world applications. For instance, in architecture, arches and domes often utilize circular shapes for their structural integrity and aesthetic appeal. Understanding the equation of a circle allows architects to precisely design these elements, ensuring they meet both functional and visual requirements. Similarly, in navigation, circles are used to define the range of a radar system or the coverage area of a radio tower, helping to map and manage signal distribution effectively.
