Which equation represents the general form 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)
Start with the standard form of a circle's equation: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( − 2 , − 3 ) and radius 4 into the equation: ( x + 2 ) 2 + ( y + 3 ) 2 = 16 .
Expand and simplify the equation to get the general form: x 2 + y 2 + 4 x + 6 y − 3 = 0 .
The equation of the circle in general form is: x 2 + y 2 + 4 x + 6 y − 3 = 0 .
Explanation
Problem Analysis The problem provides the center and diameter of a circle and asks for the equation of the circle in general form. The center is ( − 2 , − 3 ) and the diameter is 8, so the radius is r = f r a c 8 2 = 4 .
Standard Equation of a Circle The general form of a circle's equation is given by:
( x − h ) 2 + ( y − k ) 2 = r 2
where ( h , k ) is the center of the circle and r is the radius.
Substitute Values Substitute the given values h = − 2 , k = − 3 , and r = 4 into the standard equation:
( x − ( − 2 ) ) 2 + ( y − ( − 3 ) ) 2 = 4 2
( x + 2 ) 2 + ( y + 3 ) 2 = 16
Expand the Equation Expand the equation:
( x 2 + 4 x + 4 ) + ( y 2 + 6 y + 9 ) = 16
x 2 + 4 x + 4 + y 2 + 6 y + 9 = 16
Simplify to General Form Rearrange the terms to get the general form:
x 2 + y 2 + 4 x + 6 y + 4 + 9 − 16 = 0
x 2 + y 2 + 4 x + 6 y + 13 − 16 = 0
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
Therefore, the correct answer is x 2 + y 2 + 4 x + 6 y − 3 = 0 .
Examples
Understanding the equation of a circle is very useful in many real-world applications. For example, civil engineers use it to design circular structures such as tunnels and roundabouts. Architects use it to create aesthetically pleasing designs in buildings. Also, in computer graphics, the equation of a circle is used to draw circles and arcs on the screen. Knowing the center and radius, we can easily define a circular path for objects to follow in games or simulations.
