Let `A-=(-1,0),B-=(3,0),` and `P Q` be any line passing through (4, 1) having slope `mdot` Find the range of `m` for which there exist two points on `P Q` at which `A B` subtends a right angle.

Explanation

To solve the problem, we need to find the range of the slope \( m \) for which the line \( PQ \) passing through the point \( (4, 1) \) intersects the circle defined by the points \( A(-1, 0) \) and \( B(3, 0) \) at two points such that \( AB \) subtends a right angle at those points.

### Step-by-Step Solution:

1. **Find the Equation of the Circle:**
The points \( A(-1, 0) \) and \( B(3, 0) \) define the diameter of the circle. The center of the circle \( C \) is the midpoint of \( AB \):
\[
C\left(\frac{-1 + 3}{2}, \frac{0 + 0}{2}\right) = (1, 0)
...