Question

Which of the following is the equation for the circle above?

Answer 1

Recall that the general equation of a circle is given by the equation

`(x-h)^2+(y-k)^2=r^2`

where r is the radius of the circle and (h,k) is the center of the circle. So, we need to determine both the radius and the center to determine the equation.

Note that points (0,0) and (0,6) are the lowest and the highest point of the circle and that they are on a vertical line. By definition, the midpoint of the segment of this two points would be the center of the circle. Then, recall that the midpoint of the segmente that joins points (a,b) and (c,d) is given by the formula

`((a+c)/(2),(b+d)/(2))`

So, in our case, we have

`((0+0)/(2),(0+6)/(2))=(0,3)`

Thus, we have h=0 and k=3. Now we need to find the value of r. Note that r would be one half of the distance between points (0,0) and (0,6). Since this points are located on a vertical line, we can find the distance between them by subtracting the y coordinates as follows:

`|6-0|=6`

Since the distance between (0,0) and (0,6) is 6, we have that the radius would be r=3. So the final equation would be

`x^2+(y-3)^2=3^2=9`