To solve the problem, we need to analyze both statements provided and verify their correctness step by step.
### Step 1: Understanding the Parabola
The given parabola is \( y^2 = 4ax \). This is a standard form of a parabola that opens to the right, where \( a \) is the distance from the vertex to the focus.
**Hint:** Familiarize yourself with the properties of the parabola, including its focus, directrix, and the general equation.
### Step 2: Focal Chord Definition
A focal chord is a line segment that passes through the focus of the parabola and has its endpoints on the parabola itself. For the parabola \( y^2 = 4ax \), the focus is at \( (a, 0) \).
**Hint:** Remember that a focal chord connects two points on the parabola and passes through the focus.
### Step 3: Equation of Tangents
The equation of the tangent to the parabola at a point \( (at^2, 2at) \) is given by:
\[
ty = x + at^2
\]
For two points \( P(at_1^2, 2at_1) \) and \( Q(at_2^2, 2at_2) \) on the parabola, the tangents at these points are:
1. \( t_1y = x + at_1^2 \)
2. \( t_2y = x + at_2^2 \)
**Hint:** Use the tangent formula to derive the equations of the tangents at the endpoints of the focal chord.
### Step 4: Finding the Intersection of Tangents
To find the intersection of the tangents, we can set the two tangent equations equal to each other:
\[
t_1y - at_1^2 = t_2y - at_2^2
\]
Rearranging gives:
\[
y(t_1 - t_2) = a(t_1^2 - t_2^2)
\]
From this, we can solve for \( y \) and then substitute back to find \( x \).
**Hint:** Use algebraic manipulation to isolate \( y \) and then substitute to find \( x \).
### Step 5: Condition for Focal Chord
For a focal chord, the product of the slopes of the tangents is \( t_1t_2 = -1 \). This means that the tangents are perpendicular.
**Hint:** Recall that if the product of slopes is -1, the lines are perpendicular.
### Step 6: Finding the Intersection Point
The intersection point \( R \) of the tangents can be found using:
\[
x = -a, \quad y = a(t_1 + t_2)
\]
Since \( t_1t_2 = -1 \), we can express \( t_1 + t_2 \) in terms of \( t_1 \) and \( t_2 \).
**Hint:** Use the relationships derived from the properties of the focal chord to simplify your calculations.
### Step 7: Conclusion for Statement 1
The intersection point of the tangents lies on the line \( x + a = 0 \), confirming that Statement 1 is true.
### Step 8: Analyzing Statement 2
The locus of the point of intersection of perpendicular tangents to the parabola is indeed the directrix, which is given by the equation \( x = -a \).
**Hint:** Recall the definition of the directrix and its geometric significance in relation to the parabola.
### Final Conclusion
Both statements are true:
- **Statement 1:** True, as the tangents intersect on the line \( x + a = 0 \).
- **Statement 2:** True, as the locus of the intersection of perpendicular tangents is the directrix.
### Final Answer
Both statements are true, and thus the correct option is the one that states both statements are true.
