What is the equation of the circle with center $(3.2,-2.1)$ and radius $4.3$?
A. $(x-3.2)^2+(y+2.1)^2=(4.3)^2$
B. $(x+3.2)^2+(y-2.1)^2=4.3$
C. $(x-2.1)^2-(y+3.2)^2=(4.3)^2$
D. $(x+2.1)^2+(y-3.2)^2=8.6
Answer (2)
Recall the general equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( h , k ) = ( 3.2 , − 2.1 ) and radius r = 4.3 into the equation.
Simplify the equation to get ( x − 3.2 ) 2 + ( y + 2.1 ) 2 = ( 4.3 ) 2 .
The equation of the circle is ( x − 3.2 ) 2 + ( y + 2.1 ) 2 = ( 4.3 ) 2 .
Explanation
Recall the general equation of a circle. The equation of a circle with center ( h , k ) and radius r is given by the formula: ( x − h ) 2 + ( y − k ) 2 = r 2
Identify the given values. In this problem, we are given the center of the circle as ( 3.2 , − 2.1 ) and the radius as 4.3 . Therefore, we have h = 3.2 , k = − 2.1 , and r = 4.3 .
Substitute the values into the equation. Substitute the given values into the equation of the circle: ( x − 3.2 ) 2 + ( y − ( − 2.1 ) ) 2 = ( 4.3 ) 2 ( x − 3.2 ) 2 + ( y + 2.1 ) 2 = ( 4.3 ) 2
State the final answer. The equation of the circle with center ( 3.2 , − 2.1 ) and radius 4.3 is ( x − 3.2 ) 2 + ( y + 2.1 ) 2 = ( 4.3 ) 2 . Comparing this with the given options, we see that option A matches our result.
Examples
Circles are fundamental in many real-world applications. For example, in architecture, arches and domes often utilize circular shapes for their structural integrity and aesthetic appeal. In engineering, gears and wheels are based on circular designs to ensure smooth and efficient motion. Understanding the equation of a circle allows engineers and architects to precisely design and construct these structures, ensuring they meet specific requirements for size, placement, and functionality. For instance, designing a circular gear with a specific radius involves using the circle's equation to define its shape and ensure it meshes correctly with other gears.
The equation of the circle with center ( 3.2 , − 2.1 ) and radius 4.3 is given by the formula ( x − 3.2 ) 2 + ( y + 2.1 ) 2 = ( 4.3 ) 2 . Comparing this with the provided options, we find that Option A is the correct choice.
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