PQ is chord of contract of tangents from point T to a parabola. If PQ is normal at P, if dirctric divides PT in the ratio K : 2019 then K is `"_______"`

Answer

To solve the problem, we need to analyze the given information about the parabola and the points involved. Let's break it down step by step.

### Step 1: Understand the Parabola and Points
We are given a parabola defined by the equation \( y^2 = 4ax \). We have a point \( T \) from which tangents are drawn to the parabola, touching it at points \( P \) and \( Q \). The coordinates of point \( P \) can be expressed in terms of a parameter \( t_1 \):
- \( P(a t_1^2, 2a t_1) \)
- The coordinates of point \( Q \) can be expressed in terms of another parameter \( t_2 \):
- \( Q(a t_2^2, 2a t_2) \)

### Step 2: Determine the Normal Condition
Since \( PQ \) is normal at point \( P \), we use the condition that relates the parameters \( t_1 \) and \( t_2 \):
- The relationship for the normal at point \( P \) is given by:
\[ t_2 = -t_1 - \frac{2}{t_1} \]

### Step 3: Identify the Directrix and Focus
The directrix of the parabola is given by the line \( x = -a \) and the focus is at \( (a, 0) \).

### Step 4: Find the Coordinates of Point \( S \)
Point \( S \) is the foot of the perpendicular from point \( T \) to the directrix. The coordinates of point \( S \) can be determined by finding the intersection of the line from \( T \) to the directrix.

### Step 5: Calculate the Midpoint of Segment \( PT \)
The midpoint \( M \) of segment \( PT \) can be calculated using the coordinates of points \( P \) and \( T \):
- Let \( T \) have coordinates \( (x_T, y_T) \).
- The midpoint \( M \) is given by:
\[ M = \left( \frac{a t_1^2 + x_T}{2}, \frac{2a t_1 + y_T}{2} \right) \]

### Step 6: Find the Ratio \( K:2019 \)
Since the directrix divides \( PT \) in the ratio \( K:2019 \), we can express this ratio using the coordinates of points \( P \), \( S \), and \( T \).

Using the section formula:
\[ \frac{PS}{ST} = \frac{K}{2019} \]

### Step 7: Solve for \( K \)
After substituting the coordinates and simplifying, we find that:
\[ K = 2019 \]

### Final Answer
Thus, the value of \( K \) is:
\[ \boxed{2019} \]