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 (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 ( − 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
Understanding the problem The problem asks us to find the equation of a circle given its center and diameter. Let's first recall the standard form of a circle's equation.
Recalling the circle equation The general equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Substituting the center coordinates We are given the center of the circle as ( − 4 , 9 ) . Thus, h = − 4 and k = 9 . Substituting these values into the general equation, we get: ( x − ( − 4 ) ) 2 + ( y − 9 ) 2 = r 2 ( x + 4 ) 2 + ( y − 9 ) 2 = r 2
Calculating the radius We are given the diameter of the circle as 10 units. The radius r is half of the diameter, so: r = 2 d iam e t er = 2 10 = 5
Substituting the radius value Now, we substitute the value of the radius r = 5 into the equation: ( x + 4 ) 2 + ( y − 9 ) 2 = 5 2 ( x + 4 ) 2 + ( y − 9 ) 2 = 25
Identifying the correct option 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 crucial in various real-world applications. For instance, when designing a circular garden, knowing the center and radius helps determine the placement and size of the garden. Similarly, in architecture, circular arches and domes require precise calculations based on the circle's equation to ensure structural integrity. In navigation, the equation of a circle can be used to define the range of a radar system or the area covered by a radio tower. These examples highlight how the seemingly abstract equation of a circle has practical implications in design, construction, and technology.
The equation of the circle with a center at (-4, 9) and a diameter of 10 units is given by ( x + 4 ) 2 + ( y − 9 ) 2 = 25 , which corresponds to option B. This is obtained by determining the radius from the diameter and substituting into the standard form of the circle's equation.
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