A man strats from the point `P(-3,4)` and reaches the point `Q(0,1)` touching the x-axis at `R(alpha,0)` such that `PR+RQ` is minimum. Then `|alpha|=`.

Explanation

To solve the problem of finding the value of \(|\alpha|\) where a man travels from point \(P(-3, 4)\) to point \(Q(0, 1)\) touching the x-axis at point \(R(\alpha, 0)\) such that the distance \(PR + RQ\) is minimized, we can follow these steps:

### Step 1: Understand the Geometry
The man travels from point \(P\) to point \(Q\) via point \(R\) on the x-axis. The coordinates of \(R\) are \((\alpha, 0)\). We need to minimize the total distance \(PR + RQ\).

### Step 2: Set Up the Distance Formula
The distance \(PR\) can be calculated using the distance formula:
\[ PR = \sqrt{(\alpha + 3)^2 + (0 - 4)^2} = \sqrt{(\alpha + 3)^2 + 16} \]
The distance \(RQ\) is:
\[ RQ = \sqrt{(0 - \alpha)^2 + (1 - 0)^2} = \sqrt{\alpha^2 + 1} \]

### Step 3: Total Distance
The total distance \(D\) is given by:
\[ D = PR + RQ = \sqrt{(\alpha + 3)^2 + 16} + \sqrt{\alpha^2 + 1} \]

### Step 4: Use Reflection to Minimize Distance
To minimize the distance, we can reflect point \(Q\) across the x-axis to point \(Q'(0, -1)\). The straight line from \(P\) to \(Q'\) will intersect the x-axis at point \(R\).

### Step 5: Find the Equation of the Line
The slope of the line \(PQ'\) is:
\[ \text{slope} = \frac{-1 - 4}{0 + 3} = \frac{-5}{3} \]
Using point-slope form, the equation of the line through point \(P(-3, 4)\) is:
\[ y - 4 = \frac{-5}{3}(x + 3) \]
Simplifying this:
\[ y - 4 = \frac{-5}{3}x - 5 \implies y = \frac{-5}{3}x - 1 \]

### Step 6: Find the x-intercept (where \(y = 0\))
Set \(y = 0\) to find the x-intercept:
\[ 0 = \frac{-5}{3}x - 1 \implies \frac{5}{3}x = -1 \implies x = -\frac{3}{5} \]
Thus, \(\alpha = -\frac{3}{5}\).

### Step 7: Calculate \(|\alpha|\)
Now, we find \(|\alpha|\):
\[ |\alpha| = \left| -\frac{3}{5} \right| = \frac{3}{5} \]

### Final Answer
\[ |\alpha| = \frac{3}{5} \]