The equation of the line is given by y = mx - 2.

To find when the line is tangent to the curve y = x^2 - 2x + 7, equate the two equations:

\(x^2 - 2x + 7 = mx - 2\)

Rearrange to form a quadratic equation:

\(x^2 - (2 + m)x + 9 = 0\)

For the line to be tangent, the discriminant must be zero:

\(b^2 - 4ac = 0\)

Substitute into the discriminant formula:

\((2 + m)^2 - 4 \times 1 \times 9 = 0\)

Simplify and solve for m:

\((2 + m)^2 = 36\)

\(2 + m = 6 \quad \text{or} \quad 2 + m = -6\)

\(m = 4 \quad \text{or} \quad m = -8\)

For m = 4, substitute back to find x:

\(x^2 - 6x + 9 = 0\)

\((x - 3)^2 = 0\)

\(x = 3\)

Substitute x = 3 into the line equation to find y:

\(y = 4 \times 3 - 2 = 10\)

Coordinates are (3, 10).

For m = -8, substitute back to find x:

\(x^2 + 6x + 9 = 0\)

\((x + 3)^2 = 0\)

\(x = -3\)

Substitute x = -3 into the line equation to find y:

\(y = -8 \times (-3) - 2 = 22\)

Coordinates are (-3, 22).