To find the equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes, we can use the general equation of a circle:

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

where (h, k) is the center of the circle and r is the radius.

Step 1: Finding the center of the circle
Since the circle passes through (0, 0), we can substitute these coordinates into the equation:

(0 - h)^2 + (0 - k)^2 = r^2

Simplifying this equation gives us:

h^2 + k^2 = r^2 ----(1)

Step 2: Finding the radius of the circle
The circle makes intercepts 2 and 4 on the coordinate axes. This means that the distance between the center of the circle and the x-axis is equal to 2, and the distance between the center of the circle and the y-axis is equal to 4.

Using the distance formula, we can write the following equations:

√(h^2 + (k - 4)^2) = 2 ----(2)
√((h - 2)^2 + k^2) = 4 ----(3)

Squaring equation (2), we get:

h^2 + (k - 4)^2 = 4 ----(4)

Squaring equation (3), we get:

(h - 2)^2 + k^2 = 16 ----(5)

Expanding equation (4) gives us:

h^2 + k^2 - 8k + 16 = 4 ----(6)

Expanding equation (5) gives us:

h^2 - 4h + 4 + k^2 = 16 ----(7)

Combining equations (6) and (7), we get:

-4h - 8k + 4 = 0 ----(8)

Step 3: Substituting the values into the equation of the circle
Substituting the values of h^2 + k^2 from equation (1) into equation (8), we get:

r^2 - 4h - 8k + 4 = 0

Rearranging this equation, we get:

r^2 = 4h + 8k - 4 ----(9)

Substituting the values of h^2 + k^2 from equation (1) into equation (9), we get:

r^2 = 4r^2 - 4

Simplifying this equation gives us:

3r^2 = 4

Therefore, r^2 = 4/3

Step 4: Writing the final equation of the circle
Substituting the values of h^2 + k^2 and r^2 into the general equation of a circle, we get:

(x - 0)^2 + (y - 0)^2 = 4/3

Simplifying this equation gives us:

x^2 + y^2 = 4/3

So, the correct equation of the circle passing through (0, 0) and making intercepts 2 and 4 on the coordinate axes is:

x^2 + y^2 - 4/3 =