Which equation represents a circle with a center at $(-4,9)$ and a diameter of 10 units?
A. $(x-9)^2+(y+4)^2=25$
B. $(x+4)^2+(y-9)^2=25$
C. $(x-9)^2+(y+4)^2=100$
D. $(x+4)^2+(y-9)^2=100
Answer (1)
The equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( − 4 , 9 ) into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = r 2 .
Calculate the radius from the diameter: r = 2 10 = 5 .
Substitute the radius into the equation to get the final answer: ( x + 4 ) 2 + ( y − 9 ) 2 = 25 .
Explanation
Problem Analysis The problem asks us to find the equation of a circle given its center and diameter. We know the center is at ( − 4 , 9 ) and the diameter is 10 units.
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 center coordinates In our case, the center is ( h , k ) = ( − 4 , 9 ) . So, we substitute these values into the general equation: ( x − ( − 4 ) ) 2 + ( y − 9 ) 2 = r 2 ( x + 4 ) 2 + ( y − 9 ) 2 = r 2
Calculate the radius We are given the diameter of the circle as 10 units. The radius is half of the diameter, so we calculate the radius: r = 2 d iam e t er = 2 10 = 5
Substitute the radius value Now we substitute the radius r = 5 into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = 5 2 ( x + 4 ) 2 + ( y − 9 ) 2 = 25
Identify the correct equation Comparing this equation with the given options, we find that the correct equation is: ( x + 4 ) 2 + ( y − 9 ) 2 = 25
Examples
Understanding the equation of a circle is very useful in various real-world applications. For example, civil engineers use it when designing circular tunnels or roundabouts. Architects use it to design domes or arched windows. In navigation, the equation of a circle helps define the range of a radar system or the coverage area of a radio tower. By knowing the center and radius, we can precisely define and work with circular shapes in many practical scenarios.
