The tangent at one extremity of a focal chord of the parabola `y^2 = 4ax` is parallel to the normal drawn at the other extremity.

Answer

To solve the problem, we need to analyze the given conditions regarding the parabola \( y^2 = 4ax \) and the focal chord defined by points \( P \) and \( Q \).

### Step-by-Step Solution:

1. **Identify the Focus and Points on the Parabola:**
The focus of the parabola \( y^2 = 4ax \) is at the point \( S(a, 0) \). The points \( P \) and \( Q \) on the parabola can be expressed in parametric form:
- Let \( P \) be represented by \( (at_1^2, 2at_1) \)
- Let \( Q \) be represented by \( (at_2^2, 2at_2) \)

2. **Condition for Focal Chord:**
For \( P \) and \( Q \) to be endpoints of a focal chord, the relationship between their parameters must hold:
\[ t_1 \cdot t_2 = -1 \]

3. **Equation of Tangent at Point \( P \):**
The equation of the tangent line at point \( P \) is given by:
\[ t_1 y = x + at_1^2 \]
Rearranging this, we can express it in slope-intercept form:
\[ y = \frac{1}{t_1} x + at_1 \]
Thus, the slope \( m_1 \) of the tangent at \( P \) is:
\[ m_1 = \frac{1}{t_1} \]

4. **Equation of Normal at Point \( Q \):**
The equation of the normal line at point \( Q \) is given by:
\[ y = -t_2 x + 2at_2 + at_2^3 \]
Rearranging this gives us the slope \( m_2 \) of the normal at \( Q \):
\[ m_2 = -t_2 \]

5. **Setting Slopes Equal:**
According to the problem, the tangent at \( P \) is parallel to the normal at \( Q \). Therefore, we set the slopes equal:
\[ m_1 = m_2 \]
Substituting the expressions for \( m_1 \) and \( m_2 \):
\[ \frac{1}{t_1} = -t_2 \]

6. **Substituting the Condition for Focal Chord:**
From the condition \( t_1 \cdot t_2 = -1 \), we can express \( t_1 \) in terms of \( t_2 \):
\[ t_1 = -\frac{1}{t_2} \]

7. **Verification:**
Substitute \( t_1 = -\frac{1}{t_2} \) into the equation \( \frac{1}{t_1} = -t_2 \):
\[ \frac{1}{-\frac{1}{t_2}} = -t_2 \implies t_2 = -t_2 \]
This confirms that the slopes are indeed equal, validating the condition that the tangent at one extremity of the focal chord is parallel to the normal at the other extremity.

### Conclusion:
The given statement is true, and we have shown that the tangent at one extremity of a focal chord of the parabola \( y^2 = 4ax \) is parallel to the normal drawn at the other extremity.