Which equation represents a circle with a center at $(-3,-5)$ and a radius of 6 units?
A. $(x-3)^2+(y-5)^2=6$
B. $(x-3)^2+(y-5)^2=36$
C. $(x+3)^2+(y+5)^2=6$
D. $(x+3)^2+(y+5)^2=36
Answer (1)
Recall the general equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( − 3 , − 5 ) for ( h , k ) and the radius 6 for r .
Simplify the equation: ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
The equation of the circle is ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
Explanation
Understanding the problem The problem asks us to find the equation of a circle given its center and radius. We know the general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Identifying given values We are given that the center of the circle is ( − 3 , − 5 ) , so h = − 3 and k = − 5 . We are also given that the radius is 6, so r = 6 .
Substituting values into the general equation Now, we substitute these values into the general equation of a circle: ( x − ( − 3 ) ) 2 + ( y − ( − 5 ) ) 2 = 6 2 Simplifying this, we get: ( x + 3 ) 2 + ( y + 5 ) 2 = 36
Final equation Therefore, the equation of the circle with center ( − 3 , − 5 ) and radius 6 is ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
Examples
Understanding the equation of a circle is useful in many real-world applications. For example, engineers use it to design circular structures like tunnels and wheels. Architects use it to create curved designs in buildings. Even in fields like astronomy, the equation of a circle helps describe the orbits of planets around stars. Knowing how to find the equation of a circle given its center and radius allows us to model and understand these circular phenomena.
