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 will determine the conditions under which the line \( y = mx + c \) touches the parabola \( y^2 = 12(x + 3) \) exactly for one value of \( m \) (where \( m > 0 \)). We will then find the value of \( \frac{c + m}{c - m} \).

### Step 1: Rewrite the parabola equation
The given parabola is:
\[ y^2 = 12(x + 3) \]
This can be rewritten as:
\[ y^2 = 12x + 36 \]

### Step 2: Write the equation of the tangent line
The equation of the line is given by:
\[ y = mx + c \]
For this line to be tangent to the parabola, we will substitute \( y \) from the line equation into the parabola equation.

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

### Step 4: Condition for tangency
For the line to touch the parabola at exactly one point, the discriminant of this quadratic equation must be zero:
\[ (2mc - 12)^2 - 4m^2(c^2 - 36) = 0 \]

### Step 5: Expand and 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:
\[ -48mc + 144 = -144m^2 \]
Rearranging gives:
\[ 48mc = 144 + 144m^2 \]
Dividing by 48:
\[ mc = 3 + 3m^2 \]

### Step 6: Rearranging to form a quadratic in \( m \)
Rearranging gives:
\[ 3m^2 - mc + 3 = 0 \]

### Step 7: Condition for equal roots
For this quadratic in \( m \) to have equal roots, the discriminant must be zero:
\[ (-c)^2 - 4 \cdot 3 \cdot 3 = 0 \]
This simplifies to:
\[ c^2 - 36 = 0 \]
Thus:
\[ c^2 = 36 \implies c = 6 \text{ or } c = -6 \]
Since \( m > 0 \), we will consider \( c = 6 \).

### Step 8: Substitute \( c \) back to find \( m \)
Substituting \( c = 6 \) back into the equation \( mc = 3 + 3m^2 \):
\[ 6m = 3 + 3m^2 \]
Rearranging gives:
\[ 3m^2 - 6m + 3 = 0 \]
Dividing by 3:
\[ m^2 - 2m + 1 = 0 \]
Factoring gives:
\[ (m - 1)^2 = 0 \implies m = 1 \]

### Step 9: Calculate \( \frac{c + m}{c - m} \)
Now, substituting \( c = 6 \) and \( m = 1 \):
\[ \frac{c + m}{c - m} = \frac{6 + 1}{6 - 1} = \frac{7}{5} \]

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