Which of the following is the equation for a circle with a radius of $r$ and center at $(h, v)$?
A. $(x+h)^2+(y+v)^2=r^2$
B. $(x-v)^2+(y-h)^2=r^2$
C. $(x-h)^2+(y-v)^2=t^2$
D. $h^2+v^2=t^2
Answer (1)
Recall the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( h , v ) into the standard equation: ( x − h ) 2 + ( y − v ) 2 = r 2 .
Compare the resulting equation with the provided options.
Identify the correct equation: ( x − h ) 2 + ( y − v ) 2 = r 2 .
Explanation
Problem Analysis The question asks us to identify the correct equation of a circle given its center and radius. We need to recall the standard form of a circle's equation and compare it with the provided options.
Recalling the Standard Equation The standard equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2 In our case, the center is ( h , v ) , so we replace k with v in the standard equation: ( x − h ) 2 + ( y − v ) 2 = r 2 Now, we compare this equation with the given options to find the correct one.
Comparing with Options Comparing the derived equation with the given options:
A. ( x + h ) 2 + ( y + v ) 2 = r 2 - Incorrect, as it has ( x + h ) and ( y + v ) instead of ( x − h ) and ( y − v ) .
B. ( x − v ) 2 + ( y − h ) 2 = r 2 - Incorrect, as the center coordinates are swapped. C. ( x − h ) 2 + ( y − v ) 2 = t 2 - Incorrect, as it has t 2 instead of r 2 on the right side, and t is not defined. D. h 2 + v 2 = t 2 - Incorrect, as this is not the equation of a circle.
Therefore, the correct equation must be option C if we assume that t = r . However, since the problem states that the radius is r , the correct equation should have r 2 on the right side. Given the options, the closest one is C, but it should be corrected to have r 2 instead of t 2 .
Identifying the Correct Option The correct equation for a circle with radius r and center ( h , v ) is: ( x − h ) 2 + ( y − v ) 2 = r 2 Comparing this with the options, we see that option C, ( x − h ) 2 + ( y − v ) 2 = t 2 , is the closest, but it uses t 2 instead of r 2 . If we assume t = r , then option C would be correct. However, since the problem explicitly states the radius is r , the equation should have r 2 on the right side.
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, you need to know the center point and the radius to properly lay out the garden's boundaries. Similarly, in computer graphics, circles are used extensively to create various shapes and designs. Knowing the equation of a circle allows you to define its position and size accurately, whether you're designing a logo or simulating physical phenomena.
