If `y= 3x+c` is a tangent to the circle `x^2+y^2-2x-4y-5=0` , then `c` is equal to :

Options

A. `(-2, 3)`
B. `(4, -1)`
C. `(2, -3)`
D. `(4, 1)`

Explanation

To solve the problem, we need to find the value of \( c \) such that the line \( y = 3x + c \) is a tangent to the circle given by the equation \( x^2 + y^2 - 2x - 4y - 5 = 0 \).

### Step 1: Rewrite the equation of the circle in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 2x - 4y - 5 = 0 \]
We will complete the square for the \( x \) and \( y \) terms.

1. For \( x \):
\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]
2. For \( y \):
\[ y^2 - 4y \rightarrow (y - 2)^2 - 4 \]

Now substituting these back into the equation:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 - 5 = 0 \]
This simplifies to:
\[ (x - 1)^2 + (y - 2)^2 - 10 = 0 \]
Thus, we can write the equation of the circle as:
\[ (x - 1)^2 + (y - 2)^2 = 10 \]
This shows that the center of the circle is \( (1, 2) \) and the radius is \( \sqrt{10} \).

### Step 2: Find the distance from the center of the circle to the line
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For the line \( y = 3x + c \), we can rewrite it in the form \( Ax + By + C = 0 \):
\[ 3x - y + c = 0 \]
Here, \( A = 3 \), \( B = -1 \), and \( C = c \).

Now, we will calculate the distance from the center \( (1, 2) \) to this line:
\[ d = \frac{|3(1) - 1(2) + c|}{\sqrt{3^2 + (-1)^2}} = \frac{|3 - 2 + c|}{\sqrt{9 + 1}} = \frac{|1 + c|}{\sqrt{10}} \]

### Step 3: Set the distance equal to the radius
Since the line is a tangent to the circle, this distance must equal the radius \( \sqrt{10} \):
\[ \frac{|1 + c|}{\sqrt{10}} = \sqrt{10} \]

### Step 4: Solve for \( c \)
Multiplying both sides by \( \sqrt{10} \):
\[ |1 + c| = 10 \]

This gives us two cases to consider:
1. \( 1 + c = 10 \)
2. \( 1 + c = -10 \)

**Case 1:**
\[ 1 + c = 10 \implies c = 10 - 1 = 9 \]

**Case 2:**
\[ 1 + c = -10 \implies c = -10 - 1 = -11 \]

### Final Answer
Thus, the values of \( c \) are:
\[ c = 9 \quad \text{or} \quad c = -11 \]