Let `S = { (x, y |(x-3) ^(2) + (y + 42) ^(2) = 196}`
If `A = min _((x,y) in S) sqrt (x ^(2) +y ^(2)) and B = max _((x,y) in S) sqrt (x ^(2) +y ^(2)), ` then `|B-A|` is equal to

Answer

To solve the problem, we need to analyze the given circle and find the minimum and maximum distances from the origin (0,0) to any point on the circle defined by the equation:

\[ S = \{(x, y) | (x - 3)^2 + (y + 42)^2 = 196\} \]

### Step 1: Identify the center and radius of the circle

The equation of the circle can be rewritten in standard form:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center and \(r\) is the radius. From the given equation:

- The center \(C\) is at \((3, -42)\).
- The radius \(r\) is \(\sqrt{196} = 14\).

### Step 2: Calculate the distance from the origin to the center of the circle

To find the minimum and maximum distances from the origin (0,0) to the points on the circle, we first calculate the distance from the origin to the center \(C(3, -42)\):

\[ d = \sqrt{(3 - 0)^2 + (-42 - 0)^2} = \sqrt{3^2 + (-42)^2} = \sqrt{9 + 1764} = \sqrt{1773} \]

### Step 3: Determine the minimum and maximum distances

The minimum distance \(A\) from the origin to the circle is given by:

\[ A = d - r = \sqrt{1773} - 14 \]

The maximum distance \(B\) from the origin to the circle is given by:

\[ B = d + r = \sqrt{1773} + 14 \]

### Step 4: Calculate \(|B - A|\)

Now, we need to find \(|B - A|\):

\[ |B - A| = |(\sqrt{1773} + 14) - (\sqrt{1773} - 14)| \]

This simplifies to:

\[ |B - A| = |14 + 14| = |28| = 28 \]

### Final Result

Thus, the value of \(|B - A|\) is:

\[ \boxed{28} \]