The length of the intercept on the y-axis, by the circle whose diameter is the line joining the points (-4, 3) and (12, -1), is: A) 3√2 B) √13 C) 4√13 D) None of these.

To solve this question, we need to find the length of the intercept on the y-axis made by the circle whose diameter is the line joining the points (-4, 3) and (12, -1).

### Step 1: Find the center and radius of the circle
The center of the circle lies at the midpoint of the diameter. We can find the midpoint using the formula:

Midpoint = \[\left( \frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2} \right)\]

Here, the coordinates of the endpoints of the diameter are (-4, 3) and (12, -1). Let's substitute the values:

Midpoint = \[\left( \frac{{-4 + 12}}{2}, \frac{{3 + (-1)}}{2} \right)\]
= \[\left( \frac{8}{2}, \frac{2}{2} \right)\]
= (4, 1)

So, the center of the circle is (4, 1).

Next, we calculate the radius of the circle, which is half the length of the diameter. The length of the diameter is the distance between the points (-4, 3) and (12, -1). We use the distance formula:

Distance = \[\sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\]

Substitute the values:

Distance = \[\sqrt{{(12 - (-4))^2 + (-1 - 3)^2}}\]
= \[\sqrt{{(12 + 4)^2 + (-4)^2}}\]
= \[\sqrt{{16^2 + 4^2}}\]
= \[\sqrt{{256 + 16}}\]
= \[\sqrt{272}\]
= \[\sqrt{4 \times 68}\]
= \[2\sqrt{68}\]

Thus, the radius is half of this length:

Radius = \[\frac{2\sqrt{68}}{2} = \sqrt{68}\]

### Step 2: Equation of the circle
The general equation of the circle with center (h, k) and radius r is:

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

Substitute the center (4, 1) and radius \(\sqrt{68}\):

\[(x - 4)^2 + (y - 1)^2 = 68\]

### Step 3: Find the y-intercept
To find the y-intercept, set \(x = 0\) in the equation of the circle and solve for \(y\):

\[(0 - 4)^2 + (y - 1)^2 = 68\]
\[(16) + (y - 1)^2 = 68\]
\[(y - 1)^2 = 68 - 16\]
\[(y - 1)^2 = 52\]
\[\Rightarrow y - 1 = \pm \sqrt{52}\]
\[\Rightarrow y - 1 = \pm 2\sqrt{13}\]
\[\Rightarrow y = 1 \pm 2\sqrt{13}\]

Thus, the y-intercepts are \(y = 1 + 2\sqrt{13}\) and \(y = 1 - 2\sqrt{13}\).

### Step 4: Length of the y-intercept
The length of the intercept on the y-axis is the absolute difference between the two y-intercepts:

Length = \[\left| (1 + 2\sqrt{13}) - (1 - 2\sqrt{13}) \right|\]
= \[\left| 1 + 2\sqrt{13} - 1 + 2\sqrt{13} \right|\]
= \[\left| 4\sqrt{13} \right|\]
= \(4\sqrt{13}\)

Thus, the length of the intercept on the y-axis is \(4\sqrt{13}\).

### Final Answer:
The correct answer is option C) \(4\sqrt{13}\).