Let AB is the focal chord of a parabola and D and C be the foot of the perpendiculars from A and B on its directrix respectively. If CD = 6 units and area of trapezium ABCD is 36 square units, then the length (in units) of the chord AB is

Answer

To solve the problem, we need to find the length of the focal chord AB of a parabola given that CD = 6 units and the area of trapezium ABCD = 36 square units.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let \( S \) be the focus of the parabola.
- Points \( A \) and \( B \) are on the parabola, and \( D \) and \( C \) are the feet of the perpendiculars dropped from \( A \) and \( B \) to the directrix of the parabola.

2. **Using the Properties of the Parabola**:
- The distances from the focus to the points on the parabola are equal to the distances from those points to the directrix.
- Therefore, we have:
\[ AS = AD \quad \text{and} \quad BS = BC \]

3. **Area of the Trapezium**:
- The area \( A \) of trapezium \( ABCD \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times (BC + AD) \times CD \]
- Substituting the known values:
\[ 36 = \frac{1}{2} \times (BC + AD) \times 6 \]

4. **Simplifying the Area Equation**:
- Multiply both sides by 2:
\[ 72 = (BC + AD) \times 6 \]
- Divide both sides by 6:
\[ BC + AD = 12 \]

5. **Substituting the Distances**:
- Since \( BC = BS \) and \( AD = AS \), we can rewrite the equation:
\[ BS + AS = 12 \]
- From the properties of the parabola, we know that:
\[ AB = BS + AS \]
- Therefore, we can conclude:
\[ AB = 12 \]

### Final Answer:
The length of the focal chord \( AB \) is \( 12 \) units.