PQ is any focal of the parabola `y^(2)=32x`. The length of PQ cam never be less then

Options

A. 8 units
B. 16 units
C. 32 units
D. 48 units

Explanation

To solve the problem, we need to find the minimum length of the line segment PQ, where P and Q are points on the parabola defined by the equation \( y^2 = 32x \).

### Step-by-Step Solution:

1. **Identify the Parabola's Properties**:
The given equation of the parabola is \( y^2 = 32x \). This can be rewritten in the standard form \( y^2 = 4Ax \), where \( A \) is the distance from the vertex to the focus.

Comparing \( 4A = 32 \), we find:
\[ A = \frac{32}{4} = 8 \]

2. **Determine the Vertex and Focus**:
The vertex of the parabola is at the origin \( (0, 0) \). The focus of the parabola, which is located at \( (A, 0) \), is:
\[ (8, 0) \]

3. **Understanding the Length of the Latus Rectum**:
The length of the latus rectum of a parabola is given by \( 4A \). Since we have already calculated \( A = 8 \), we can find the length of the latus rectum:
\[ \text{Length of Latus Rectum} = 4A = 4 \times 8 = 32 \]

4. **Finding Points on the Latus Rectum**:
The endpoints of the latus rectum can be found by substituting \( x = 8 \) into the parabola equation:
\[ y^2 = 32 \times 8 = 256 \implies y = \pm 16 \]
Thus, the points on the latus rectum are \( (8, 16) \) and \( (8, -16) \).

5. **Calculating the Length of PQ**:
The length of the segment PQ, which is the distance between the points \( (8, 16) \) and \( (8, -16) \), can be calculated using the distance formula:
\[ \text{Distance} = \sqrt{(8 - 8)^2 + (16 - (-16))^2} = \sqrt{0 + (16 + 16)^2} = \sqrt{32^2} = 32 \]

6. **Conclusion**:
Therefore, the length of PQ can never be less than \( 32 \).

### Final Answer:
The length of PQ can never be less than **32 units**.