Find the equation of the circle on the y-axis and has y-intercept at 1 and 7.
the center of the circle is the average of the intercepts ... (1 - 7) / 2 = 4
... center at (0,4)
the radius is the distance from the center to one of the intercepts ... 7 - 4 = 3
the general circle equation is
... (x - h)^2 + (y - k)^2 = r^2
... center at (h,k) , radius = r
To find the equation of a circle on the y-axis with the y-intercepts at (0,1) and (0,7), we need to determine the center and radius of the circle.
Since the circle is on the y-axis, its center will lie on the x-axis and have a y-coordinate equal to the average of the y-intercepts. Therefore, the center of the circle is (0, (1+7)/2) = (0, 4).
To calculate the radius of the circle, we need to find the distance between the center and any point on the circle. In this case, we can use the distance formula.
The distance formula for two points (x1, y1) and (x2, y2) is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Considering the center as (0, 4) and one of the y-intercepts as (0, 1), we can calculate the radius as:
radius = sqrt((0 - 0)^2 + (1 - 4)^2)
= sqrt(0 + 9)
= sqrt(9)
= 3
Now that we have the center (0, 4) and radius 3, we can write the equation of the circle in the standard form:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values, the equation of the circle becomes:
(x - 0)^2 + (y - 4)^2 = 3^2
Simplifying further:
x^2 + (y - 4)^2 = 9
Therefore, the equation of the circle on the y-axis with y-intercepts at 1 and 7 is:
x^2 + (y - 4)^2 = 9