If `y=m x+c` touches the parabola `y^2=4a(x+a),` then (a)`c=a/m` (b) `c=a m+a/m` `c=a+a/m` (d) none of these

Options

A. `c=(1)/(m)`
B. `c=am+(a)/(m)`
C. `c=a+(a)/(m)`
D. none of these

Explanation

To solve the problem, we need to determine the relationship between the constants \( m \), \( c \), and \( a \) when the line \( y = mx + c \) touches the parabola \( y^2 = 4a(x + a) \).

### Step-by-Step Solution:

1. **Identify the Parabola and its Tangent Equation:**
The given parabola is \( y^2 = 4a(x + a) \). We can rewrite this in the standard form of a parabola that opens to the right. The vertex of this parabola is at \( (-a, 0) \).

2. **Equation of the Tangent to the Parabola:**
The general equation of the tangent to the parabola \( y^2 = 4ax \) at a point \( (x_0, y_0) \) is given by:
\[ yy_0 = 2a(x + x_0) \]
For our parabola, we can derive the tangent line in the form \( y = mx + \frac{a}{m} \) where \( m \) is the slope of the tangent.

3. **Comparing the Tangent Line with the Given Line:**
We are given the line \( y = mx + c \). For the line to be tangent to the parabola, it must match the form of the tangent we derived. Thus, we equate:
\[ c = \frac{a}{m} + am \]

4. **Rearranging the Equation:**
Rearranging the equation gives us:
\[ c = am + \frac{a}{m} \]

5. **Conclusion:**
Therefore, the correct relationship between \( c \), \( a \), and \( m \) is:
\[ c = am + \frac{a}{m} \]

### Answer:
The correct option is (b) \( c = am + \frac{a}{m} \).