Find the equation of the family of circles touching the lines `x^2-y^2+2y-1=0.`

Answer

To find the equation of the family of circles touching the lines given by the equation \(x^2 - y^2 + 2y - 1 = 0\), we will follow these steps:

### Step 1: Factor the given equation
The first step is to factor the equation \(x^2 - y^2 + 2y - 1 = 0\).

\[ x^2 - (y^2 - 2y + 1) = 0 \]
\[ x^2 - (y - 1)^2 = 0 \]

This can be factored as:
\[ (x - (y - 1))(x + (y - 1)) = 0 \]

Thus, we have two lines:
1. \(x - y + 1 = 0\) (or \(x - y = -1\))
2. \(x + y - 1 = 0\) (or \(x + y = 1\))