What is the equation of a circle with center $(-2,3)$ and radius $4$?
A. $(x+2)^2-(y-3)^2=16$
B. $(x+2)^2+(y-3)^2=16$
C. $(x+2)^2+(y-3)^2=4$
D. $(x-2)^2+(y+3)^2=16
Answer (2)
Recall 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.
Substitute the given center ( − 2 , 3 ) for ( h , k ) and the radius 4 for r into the equation.
Simplify the equation: ( x + 2 ) 2 + ( y − 3 ) 2 = 16 .
The equation of the circle is ( x + 2 ) 2 + ( y − 3 ) 2 = 16 .
Explanation
Problem Analysis The problem asks for the equation of a circle given its center and radius. The center is ( − 2 , 3 ) and the radius is 4 .
Recall the General Equation of a Circle The general equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Substitute the Values Substitute the given values h = − 2 , k = 3 , and r = 4 into the general equation: ( x − ( − 2 ) ) 2 + ( y − 3 ) 2 = 4 2
Simplify the Equation Simplify the equation: ( x + 2 ) 2 + ( y − 3 ) 2 = 16
Final Answer Comparing the result with the given options, we find that option B matches the equation we derived. Therefore, the equation of the circle is ( x + 2 ) 2 + ( y − 3 ) 2 = 16 .
Examples
Understanding the equation of a circle is very useful in various real-world applications. For example, civil engineers use it to design circular structures like tunnels and roundabouts. Architects use it to create aesthetically pleasing designs in buildings. In computer graphics, it's used to draw circles and arcs on the screen. Even in astronomy, the equation of a circle helps to describe the orbits of planets around stars, assuming they are perfectly circular. The equation of a circle is a fundamental concept with far-reaching applications.
The equation of the circle with center ( − 2 , 3 ) and radius 4 is given by ( x + 2 ) 2 + ( y − 3 ) 2 = 16 . This matches option B from the choices provided. Thus, the correct answer is option B.
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