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 problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Understand the Setup
We have a parabola with focus \( S \). A 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 \( PQ = PS + SQ = 3x \).

**Hint:** Visualize the parabola and mark the points \( P \), \( Q \), and \( S \) clearly.

### Step 2: Identify the Tangent and Directrix
The tangent at point \( Q \) intersects the directrix at point \( R \). We know that the distance \( RQ = 6 \).

**Hint:** Draw the tangent line at point \( Q \) and mark the intersection point with the directrix as \( R \).

### Step 3: Use Similar Triangles
Since \( RQ \) is perpendicular to the directrix, we can form two triangles:
- Triangle \( PRQ \)
- Triangle \( PAS \)

Both triangles share the angle \( P \) and have a right angle at \( R \) and \( A \) respectively.

**Hint:** Label the angles clearly to show that \( \angle PRQ = \angle PAS = 90^\circ \) and \( \angle P \) is common.

### Step 4: Set Up the Proportions
From the similarity of triangles, we have:
\[ \frac{PS}{PQ} = \frac{AS}{RQ} \]
Where:
- \( PS = 2x \)
- \( SQ = x \)
- \( PQ = 3x \)
- \( RQ = 6 \)

**Hint:** Write down the known values in the proportion to set up the equation.

### Step 5: Substitute Known Values
Substituting the known values into the proportion:
\[ \frac{2x}{3x} = \frac{AS}{6} \]
This simplifies to:
\[ \frac{2}{3} = \frac{AS}{6} \]

**Hint:** Cross-multiply to solve for \( AS \).

### Step 6: Solve for \( AS \)
Cross-multiplying gives:
\[ 2 \cdot 6 = 3 \cdot AS \implies 12 = 3 \cdot AS \implies AS = \frac{12}{3} = 4 \]

**Hint:** Ensure that you simplify correctly to find \( AS \).

### Step 7: Conclusion
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.