Let the focus (S) of a parabola divides its one of the focal chords PQ in the ratio `2:1`. If the tangent at Q cuts the directrix at R such that RQ = 6, then the distance (in units) of the focus from the tangent at P is

Answer

To solve the given problem step by step, we will follow the reasoning presented in the video transcript and apply some geometric properties of parabolas.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the focus of the parabola be denoted as \( S \).
- The focal chord \( PQ \) is divided by the focus \( S \) in the ratio \( 2:1 \). This means that if \( PS = 2x \) and \( SQ = x \), then the total length \( PQ = PS + SQ = 3x \).

2. **Tangent at Point Q**:
- The tangent line at point \( Q \) intersects the directrix at point \( R \).
- We are given that \( RQ = 6 \).

3. **Analyzing the Triangles**:
- In triangle \( PRQ \), angle \( PRQ \) is \( 90^\circ \) because \( R \) lies on the directrix and the tangent at \( Q \) is perpendicular to the line segment from \( Q \) to the directrix.
- Triangle \( PAS \) also has angle \( PAS = 90^\circ \) since \( S \) is the focus of the parabola.

4. **Using Similar Triangles**:
- Since both triangles \( PRQ \) and \( PAS \) have a right angle and share angle \( P \), they are similar by AA (Angle-Angle) similarity criterion.
- Therefore, we can write the ratio of corresponding sides:
\[ \frac{PQ}{AS} = \frac{RQ}{PS} \]

5. **Substituting Known Values**:
- We know \( RQ = 6 \) and we have established that \( PS = 2x \) and \( SQ = x \), leading to \( PQ = 3x \).
- Thus, we can express the ratio as:
\[ \frac{3x}{AS} = \frac{6}{2x} \]

6. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 3x \cdot 2x = 6 \cdot AS \]
\[ 6x^2 = 6 \cdot AS \]
- Dividing both sides by 6, we get:
\[ x^2 = AS \]

7. **Finding AS**:
- Since we know \( PS = 2x \) and \( SQ = x \), we can express \( AS \) in terms of \( RQ \):
\[ AS = \frac{2}{3} \cdot RQ = \frac{2}{3} \cdot 6 = 4 \]

8. **Conclusion**:
- Therefore, the distance of the focus \( S \) from the tangent at point \( P \) is \( AS = 4 \) units.

### Final Answer:
The distance of the focus from the tangent at \( P \) is **4 units**.