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 \((3x + c)^2\):

\[ (3x + c)^2 = 9x^2 + 6cx + c^2 \]

Substituting this back into the equation gives:

\[ 16x^2 + 9x^2 + 6cx + c^2 = 16 \]

### Step 3: Combine like terms
Combine the \(x^2\) terms:

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

This simplifies to:

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

### Step 4: Use the condition for tangency
For the line to touch the ellipse, the quadratic equation must have exactly one solution. This occurs when the discriminant \(D\) is zero.

The discriminant \(D\) for the quadratic \(Ax^2 + Bx + C = 0\) is given by:

\[ D = B^2 - 4AC \]

In our case, \(A = 25\), \(B = 6c\), and \(C = c^2 - 16\). Thus, we have:

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

### Step 5: Set the discriminant to zero
Setting the discriminant to zero gives:

\[ 36c^2 - 100(c^2 - 16) = 0 \]

### Step 6: Simplify the equation
Expanding this gives:

\[ 36c^2 - 100c^2 + 1600 = 0 \]

Combine like terms:

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

### Step 7: Solve for \(c^2\)
Rearranging gives:

\[ 64c^2 = 1600 \]

Dividing both sides by 64:

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

### Step 8: Find \(|c|\)
Taking the square root of both sides gives:

\[ |c| = 5 \]

Thus, the value of \(|c|\) for which the line touches the ellipse is:

\[ \boxed{5} \]