Find an equation of the circle that satisfies the given conditions:
Center $(4,-5); \quad$ radius 4
Answer (1)
We are given the center ( 4 , − 5 ) and radius 4 of a circle.
The standard equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 .
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
Analyze the problem and data. 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.
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 given values. Substitute the given values h = 4 , k = − 5 , and r = 4 into the standard equation: ( x − 4 ) 2 + ( y − ( − 5 ) ) 2 = 4 2
Simplify the equation. Simplify the equation: ( x − 4 ) 2 + ( y + 5 ) 2 = 16
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in GPS technology, your location is determined by finding the intersection of circles from multiple satellites. Each satellite sends a signal indicating your distance from it (the radius), and your GPS device solves for the point where these circles intersect, pinpointing your exact location on Earth. This principle extends to fields like astronomy, where the orbits of planets and stars can be modeled using circular or elliptical equations.
