To find the intercepts cut off by the circle on the x-axis and y-axis, we can use the distance formula.

The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by the equation:
d = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, we have the center of the circle passing through the origin as (3,4). The origin is (0,0).

Intercept on the x-axis:
To find the intercept on the x-axis, we need to find the distance between the center of the circle (3,4) and a point on the x-axis. Since the x-coordinate of any point on the x-axis is 0, the point on the x-axis can be represented as (0, y).

Using the distance formula, we can calculate the distance between (3,4) and (0, y):
d = √((0 - 3)^2 + (y - 4)^2)
d = √(9 + (y - 4)^2)
d = √(9 + y^2 - 8y + 16)
d = √(y^2 - 8y + 25)

Since this is the distance between two points, it is also the radius of the circle. We know that the center of the circle is (3,4), so the equation of the circle can be written as:
(x - 3)^2 + (y - 4)^2 = (y^2 - 8y + 25)

We can substitute the x-coordinate of the intercept on the x-axis, which is 0, into the equation:
(0 - 3)^2 + (y - 4)^2 = (y^2 - 8y + 25)
9 + (y - 4)^2 = (y^2 - 8y + 25)
(y - 4)^2 = y^2 - 8y + 16
y^2 - 8y + 16 - (y^2 - 8y + 25) = 0
y^2 - 8y + 16 - y^2 + 8y - 25 = 0
-9 = 0

This is a contradiction, which means that there is no intercept on the x-axis. Therefore, the intercept cut off by the circle on the x-axis is 0 units.

Intercept on the y-axis:
Similarly, to find the intercept on the y-axis, we need to find the distance between the center of the circle (3,4) and a point on the y-axis. Since the y-coordinate of any point on the y-axis is 0, the point on the y-axis can be represented as (x, 0).

Using the distance formula, we can calculate the distance between (3,4) and (x, 0):
d = √((x - 3)^2 + (0 - 4)^2)
d = √((x - 3)^2 + 16)
d = √(x^2 - 6x + 9 + 16)
d = √(x^2 - 6x + 25)

Again, since this is the distance between two