Statement-1: `y+b=m_(1) (x+a)` and `y+b=m_(2)(x+a)` are perpendicular tangents to the parabola `y^(2)=4ax`.
Statement-2: The locus of the point of intersection of perpendicular tangents to a parabola is its directrix.

Options

A. Statement-1 is True, Statement - 2 is true, Statement-2 is a correct explanation for Statement-1`
B. Statement-1 is True, Statement - 2 is true, Statement-2 is not a correct explanation for Statement-6
C. Statement-1 is True, Statement - 2 is False.
D. Statement-1 is True, Statement - 2 is True.

Explanation

To solve the given question, we will analyze both statements regarding the parabola \( y^2 = 4ax \).

### Step 1: Understand the equations of the tangents
The equations of the tangents are given as:
1. \( y + b = m_1 (x + a) \)
2. \( y + b = m_2 (x + a) \)

These lines represent tangents to the parabola \( y^2 = 4ax \).

### Step 2: Find the point of intersection
Both tangents pass through the same point, which can be found by setting the two equations equal to each other:
\[ m_1 (x + a) + b = m_2 (x + a) + b \]
This simplifies to:
\[ m_1 (x + a) = m_2 (x + a) \]
Assuming \( m_1 \neq m_2 \), we can conclude that both tangents intersect at the point:
\[ P(-a, -b) \]

### Step 3: Determine the coordinates of the point of intersection
The coordinates of the intersection point \( P \) are:
\[ P(-a, -b) \]
This means that the x-coordinate of the point of intersection is \( -a \).

### Step 4: Identify the directrix of the parabola
For the parabola \( y^2 = 4ax \), the directrix is given by the equation:
\[ x = -a \]
This indicates that the x-coordinate of the point of intersection \( P \) lies on the directrix.

### Step 5: Conclusion about the statements
- **Statement 1**: The tangents are perpendicular and intersect at the point \( P(-a, -b) \), which lies on the directrix \( x = -a \). Therefore, Statement 1 is true.
- **Statement 2**: The locus of the point of intersection of perpendicular tangents to a parabola is indeed its directrix. Thus, Statement 2 is also true.

### Final Answer
Both statements are true, and Statement 2 provides the reason for Statement 1.

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