To write the equation of a circle in standard form, we use the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Since the circle is centered at the origin, the values of h and k are both 0. Thus, our equation will simplify to:
(x - 0)^2 + (y - 0)^2 = r^2
x^2 + y^2 = r^2
To determine the radius r, we need to use the distance formula to find the distance between the center of the circle at the origin (0, 0) and the point (-2, 6) which lies on the circle. The distance formula is:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Here, (x1, y1) is the center of the circle, and (x2, y2) is the point the circle passes through.
By plugging in the values:
r = sqrt((-2 - 0)^2 + (6 - 0)^2)
r = sqrt((-2)^2 + (6)^2)
r = sqrt(4 + 36)
r = sqrt(40)
r = sqrt(4 * 10)
r = 2 * sqrt(10) (We can pull out a square factor of 4 out of the square root as 2.)
Now, we have found the radius, which is 2 * sqrt(10).
The standard form of the circle's equation with this radius is:
x^2 + y^2 = (2 * sqrt(10))^2
When we square the radius, we square both the 2 and the sqrt(10):
x^2 + y^2 = (2)^2 * (sqrt(10))^2
x^2 + y^2 = 4 * 10
x^2 + y^2 = 40
The standard form equation for the circle passing through (-2, 6) and centered at the origin is therefore:
x^2 + y^2 = 40
