If `y=m_1x+c` and `y=m_2x+c` are two tangents to the parabola `y^2+4a(x+c)=0` , then `m_1+m_2=0` (b) `1+m_1+m_2=0` `m_1m_2-1=0` (d) `1+m_1m_2=0`

Options

A. `m_(1)+m_(2)=0`
B. `1+m_(1)+m_(2)=0`
C. `m_(1)m_(2)-1=0`
D. `1+m_(1)m_(2)=0`

Explanation

To solve the problem, we need to analyze the given tangents to the parabola and derive the relationship between their slopes.

### Step-by-Step Solution:

1. **Identify the Parabola**: The given parabola is represented by the equation:
\[ y^2 + 4a(x + c) = 0 \]
This can be rewritten as:
\[ y^2 = -4a(x + c) \]
This indicates that the parabola opens to the left.

2. **Determine the Point of Intersection**: The tangents given are:
\[ y = m_1x + c \]
\[ y = m_2x + c \]
Both tangents intersect at the point \((0, c)\).

3. **Substitute the Point into the Parabola Equation**: Since the point \((0, c)\) lies on the parabola, we substitute \(x = 0\) and \(y = c\) into the parabola's equation:
\[ c^2 + 4a(0 + c) = 0 \]
This simplifies to:
\[ c^2 + 4ac = 0 \]
Factoring gives:
\[ c(c + 4a) = 0 \]
Thus, either \(c = 0\) or \(c = -4a\).

4. **Condition for Tangents**: For two tangents to a parabola that intersect at the same point, they must be perpendicular if the slopes \(m_1\) and \(m_2\) satisfy:
\[ m_1 m_2 = -1 \]
This means that the product of the slopes of the tangents is negative one.

5. **Conclusion**: The relationship between the slopes \(m_1\) and \(m_2\) is:
\[ m_1 m_2 + 1 = 0 \]
Hence, the correct option is:
\[ \text{(c) } m_1 m_2 - 1 = 0 \]