To find the equation of the circle formed by the intersection of the sphere and the plane, we can first find the equation of the intersection curve in terms of two variables, and then convert it into the standard form of a circle.

The equation of the plane is given as 2x - y + 3z = 5.

Substituting x = x, y = y, and z = (5 - 2x + y)/3 into the equation of the sphere, we get:
(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 4^2.

Expanding and simplifying the equation, we have:
x^2 - 2x + 1 + y^2 - 4y + 4 + z^2 - 6z + 9 = 16.

Rearranging the terms, we get:
x^2 + y^2 + z^2 - 2x - 4y - 6z - 12 = 0.

Now, let's express y in terms of x and z using the equation of the plane:
2x - y + 3z = 5.
y = 2x + 3z - 5.

Substituting y = 2x + 3z - 5 into the equation of the sphere, we have:
x^2 + (2x + 3z - 5)^2 + z^2 - 2x - 4(2x + 3z - 5) - 6z - 12 = 0.

Expanding and simplifying, we get:
x^2 + 4x^2 + 9z^2 + 25 - 20x - 6z + 6x + 20z - 8 - 6z - 12 = 0.

Combining like terms, we have:
5x^2 + 3z^2 - 20x + 20z - 55 = 0.

To convert this equation into the standard form of a circle, we need to complete the square for both x and z.

Rearranging the equation, we group the x-terms and z-terms separately:
(5x^2 - 20x) + (3z^2 + 20z) = 55.

Completing the square for x, we add and subtract (20/2)^2 = 100 to the x-terms inside the parentheses:
5(x^2 - 4x + 4) + (3z^2 + 20z) = 55 + 5(4).

Simplifying, we have:
5(x - 2)^2 + (3z^2 + 20z) = 75.

Completing the square for z, we add and subtract (20/2)^2 = 100 to the z-terms inside the parentheses:
5(x - 2)^2 + 3(z^2 + 20z + 100) = 75 + 3(100).

Simplifying, we get:
5(x - 2)^2 + 3(z + 10)^2 = 375.

Dividing both sides of the equation by 375 to make the coefficient of the squared terms 1, we have:
(x - 2)^2/75 + (z + 10)^2/125 = 1.

Therefore, the equation of the circle formed by the intersection of the sphere and the plane is:
(x - 2)^2/75 + (z + 10)^2/125 = 1.

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