Prove that the line `y=m(x-1)+3sqrt(1+m^(2))-2` touches the circle `x^(2)+y^(2)-2x+4y-4=0` for all real values of m.

Explanation

To prove that the line \( y = m(x - 1) + 3\sqrt{1 + m^2} - 2 \) touches the circle \( x^2 + y^2 - 2x + 4y - 4 = 0 \) for all real values of \( m \), we will follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we need to rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 2x + 4y - 4 = 0 \]
We can rearrange this as:
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