Which equation represents the general form of a circle with a center at $(-2,-3)$ and a diameter of 8 units?
A. $x^2+y^2+4 x+6 y-51=0$
B. $x^2+y^2-4 x-6 y-51=0$
C. $x^2+y^2+4 x+6 y-3=0$
D. $x^2+y^2-4 x-6 y-3=0
Answer (1)
Determine the radius from the given diameter: r = 2 8 = 4 .
Substitute the center ( − 2 , − 3 ) and radius 4 into the standard circle equation: ( x + 2 ) 2 + ( y + 3 ) 2 = 16 .
Expand and rearrange the equation to general form: x 2 + y 2 + 4 x + 6 y + 13 = 16 .
Simplify to obtain the final equation: x 2 + y 2 + 4 x + 6 y − 3 = 0 .
Explanation
Problem Analysis and Given Data The problem provides the center and diameter of a circle and asks for the equation of the circle in general form. The center is given as ( − 2 , − 3 ) and the diameter is 8, so the radius is r = 2 8 = 4 . The general form of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitute Values into Standard Equation Substitute the given center ( − 2 , − 3 ) and radius 4 into the standard equation of a circle: ( x − ( − 2 ) ) 2 + ( y − ( − 3 ) ) 2 = 4 2 Simplify the equation: ( x + 2 ) 2 + ( y + 3 ) 2 = 16
Expand the Equation Expand the equation: x 2 + 4 x + 4 + y 2 + 6 y + 9 = 16 Rearrange the terms to get the general form: x 2 + y 2 + 4 x + 6 y + 13 = 16
Simplify to General Form Subtract 16 from both sides to set the equation to zero: x 2 + y 2 + 4 x + 6 y + 13 − 16 = 0 Simplify to get the final general form equation: x 2 + y 2 + 4 x + 6 y − 3 = 0
Final Answer The equation of the circle in general form is: x 2 + y 2 + 4 x + 6 y − 3 = 0 This matches option 3.
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, consider designing a circular garden or a roundabout. Knowing the center and radius (or diameter) allows you to define the boundaries accurately using the circle's equation. This ensures precise planning and construction, whether you're laying out flower beds or determining the path of vehicles around the roundabout.
