If `y+b=m_1(x+a)`and `y+b=m_2(x+a)` are two tangents to the paraabola `y^2=4ax` then

Options

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

Answer

To solve the problem, we need to analyze the given equations of the tangents to the parabola \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Identify the Tangent Equations**:
The equations of the tangents are given as:
\[ y + b = m_1(x + a) \quad \text{(1)} \]
\[ y + b = m_2(x + a) \quad \text{(2)} \]

2. **Rearranging the Tangent Equations**:
We can rearrange both equations to express \( y \):
\[ y = m_1(x + a) - b \quad \text{(from equation 1)} \]
\[ y = m_2(x + a) - b \quad \text{(from equation 2)} \]

3. **Condition for Tangents to a Parabola**:
For these lines to be tangents to the parabola \( y^2 = 4ax \), the slopes \( m_1 \) and \( m_2 \) must satisfy a specific condition. The product of the slopes of two perpendicular lines is \(-1\):
\[ m_1 m_2 = -1 \]

4. **Conclusion**:
Since the tangents are perpendicular to each other, we conclude:
\[ m_1 m_2 = -1 \]

### Final Answer:
The product of the slopes of the two tangents \( m_1 \) and \( m_2 \) is:
\[ m_1 m_2 = -1 \]