The value of `|C|` for which the line `y=3x+c` touches the ellipse `16x^2+y^2=16` is:

Answer

To find the value of \(|c|\) for which the line \(y = 3x + c\) touches the ellipse \(16x^2 + y^2 = 16\), we can follow these steps:

### Step 1: Substitute the line equation into the ellipse equation
We start by substituting \(y = 3x + c\) into the ellipse equation \(16x^2 + y^2 = 16\).

\[ 16x^2 + (3x + c)^2 = 16 \]

### Step 2: Expand the equation
Now we expand the equation:

\[ 16x^2 + (3x + c)^2 = 16 \]
\[ 16x^2 + (9x^2 + 6cx + c^2) = 16 \]
\[ (16x^2 + 9x^2 + 6cx + c^2) = 16 \]
\[ 25x^2 + 6cx + (c^2 - 16) = 0 \]

### Step 3: Set up the discriminant condition
For the line to touch the ellipse, the discriminant of the quadratic equation must be zero. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a = 25\), \(b = 6c\), and \(c = c^2 - 16\).

The discriminant \(D\) is given by:

\[ D = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ D = (6c)^2 - 4(25)(c^2 - 16) \]

### Step 4: Simplify the discriminant
Now we simplify the discriminant:

\[ D = 36c^2 - 100(c^2 - 16) \]
\[ D = 36c^2 - 100c^2 + 1600 \]
\[ D = -64c^2 + 1600 \]

### Step 5: Set the discriminant to zero
Since we want the line to touch the ellipse, we set the discriminant \(D\) to zero:

\[ -64c^2 + 1600 = 0 \]

### Step 6: Solve for \(c^2\)
Rearranging gives:

\[ 64c^2 = 1600 \]
\[ c^2 = \frac{1600}{64} \]
\[ c^2 = 25 \]

### Step 7: Find \(|c|\)
Taking the square root of both sides gives:

\[ |c| = 5 \]

### Final Answer
The value of \(|c|\) for which the line \(y = 3x + c\) touches the ellipse \(16x^2 + y^2 = 16\) is:

\[ \boxed{5} \]