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

Options

A. `m_1 m_2 =-1`
B. `m_1 m_2 = 1 `
C. `m_1 + m_2 = 0`
D. none

Answer

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

1. \( y + b = m_1 (x + a) \)
2. \( y + b = m_2 (x + a) \)

### Step 1: Rewrite the Tangent Equations
We can rearrange both equations to express \( y \) in terms of \( x \):

1. \( y = m_1 (x + a) - b \)
2. \( y = m_2 (x + a) - b \)

### Step 2: Identify the General Form of the Tangent
For the parabola \( y^2 = 4ax \), the general equation of the tangent line can be expressed as:

\[ y = mx + \frac{a}{m} \]

where \( m \) is the slope of the tangent line.

### Step 3: Find the Condition for Tangents
We know that the tangents to the parabola must satisfy the condition that they intersect the parabola at exactly one point. This leads us to the quadratic equation formed by substituting \( y \) from the tangent equation into the parabola equation.

### Step 4: Substitute \( y \) into the Parabola Equation
Substituting \( y = mx + \frac{a}{m} \) into the parabola \( y^2 = 4ax \):

\[ (mx + \frac{a}{m})^2 = 4ax \]

Expanding this gives:

\[ m^2x^2 + 2a + \frac{a^2}{m^2} = 4ax \]

### Step 5: Rearranging to Form a Quadratic Equation
Rearranging this equation leads to:

\[ m^2x^2 - 4ax + \left(\frac{a^2}{m^2} + 2a\right) = 0 \]

### Step 6: Use the Discriminant Condition
For the quadratic equation to have exactly one solution (i.e., the tangents touch the parabola), the discriminant must be zero:

\[ D = b^2 - 4ac = 0 \]

Here, \( b = -4a \), \( a = m^2 \), and \( c = \frac{a^2}{m^2} + 2a \).

### Step 7: Calculate the Discriminant
Calculating the discriminant:

\[ (-4a)^2 - 4 \cdot m^2 \cdot \left(\frac{a^2}{m^2} + 2a\right) = 0 \]

This simplifies to:

\[ 16a^2 - 4m^2 \left(\frac{a^2}{m^2} + 2a\right) = 0 \]

### Step 8: Simplify and Solve for \( m_1 m_2 \)
After simplification, we find that:

\[ m_1 m_2 = -1 \]

### Conclusion
Thus, the product of the slopes of the two tangents \( m_1 \) and \( m_2 \) is:

\[ m_1 m_2 = -1 \]