Equate a circle with center at point T (-1; 4) and passing through point A (3; 5).

First, let’s find the radius of this circle, equal to the distance from point T with coordinates (-1; 4) to point A with coordinates (3; 5).

Using the formula for the distance between two points on the coordinate plane, we find the distance between points T and A:

| TA | = √ ((3 – (-1)) ^ 2 + (5 – 4) ^ 2) = √ ((3 + 1) ^ 2 + (5 – 4) ^ 2) = √ (4 ^ 2 + 1 ^ 2) = √ (16 + 1) = √17.

Knowing the radius of a given circle and the coordinates of its center, we can write down the equation of this circle:

(x + 1) ^ 2 + (y – 4) ^ 2 = (√17) ^ 2,

or

(x + 1) ^ 2 + (y – 4) ^ 2 = 17.

Answer: (x + 1) ^ 2 + (y – 4) ^ 2 = 17.