If the line `y=mx+c` touches the parabola `y^(2)=12(x+3)` exactly for one value of `m(m gt0)`, then the value of `(c+m)/(c-m)` is equal to

Answer

To solve the problem step by step, we need to find the value of \((c+m)/(c-m)\) given that the line \(y = mx + c\) touches the parabola \(y^2 = 12(x + 3)\) for exactly one value of \(m\) (where \(m > 0\)).

### Step 1: Set up the equations
The line is given by:
\[ y = mx + c \]
The parabola is given by:
\[ y^2 = 12(x + 3) \]

### Step 2: Substitute the line equation into the parabola equation
Substituting \(y = mx + c\) into the parabola equation:
\[ (mx + c)^2 = 12(x + 3) \]
Expanding this gives:
\[ m^2x^2 + 2mcx + c^2 = 12x + 36 \]
Rearranging the equation:
\[ m^2x^2 + (2mc - 12)x + (c^2 - 36) = 0 \]

### Step 3: Condition for tangency
For the line to be tangent to the parabola, the quadratic equation must have exactly one solution, which means the discriminant must be zero:
\[ D = (2mc - 12)^2 - 4m^2(c^2 - 36) = 0 \]

### Step 4: Simplify the discriminant
Expanding the discriminant:
\[ (2mc - 12)^2 = 4m^2(c^2 - 36) \]
This simplifies to:
\[ 4m^2c^2 - 48mc + 144 = 4m^2c^2 - 144m^2 \]
Cancelling \(4m^2c^2\) from both sides gives:
\[ -48mc + 144 + 144m^2 = 0 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 48mc = 144 + 144m^2 \]
Dividing through by 48:
\[ mc = 3 + 3m^2 \]
Thus, we have:
\[ mc - 3m^2 - 3 = 0 \]

### Step 6: Discriminant of the new quadratic
Now, we consider this as a quadratic in \(m\):
\[ 3m^2 - mc + 3 = 0 \]
For this quadratic to have exactly one solution, its discriminant must also be zero:
\[ D' = c^2 - 4 \cdot 3 \cdot 3 = 0 \]
This simplifies to:
\[ c^2 - 36 = 0 \]
Thus:
\[ c^2 = 36 \implies c = 6 \text{ or } c = -6 \]
Since \(m > 0\), we will take \(c = 6\).

### Step 7: Find the value of \(m\)
Substituting \(c = 6\) back into the equation \(mc = 3 + 3m^2\):
\[ 6m = 3 + 3m^2 \]
Rearranging gives:
\[ 3m^2 - 6m + 3 = 0 \]
Dividing through by 3:
\[ m^2 - 2m + 1 = 0 \]
This factors to:
\[ (m - 1)^2 = 0 \implies m = 1 \]

### Step 8: Calculate \((c + m)/(c - m)\)
Now we can substitute \(c = 6\) and \(m = 1\) into the expression:
\[ \frac{c + m}{c - m} = \frac{6 + 1}{6 - 1} = \frac{7}{5} \]

### Final Answer
Thus, the value of \(\frac{c + m}{c - m}\) is:
\[ \frac{7}{5} \]