If a and b be the segments of a focal chord and 4c be the latus rectum of a parabola. Then

Options

A. `a ^(2) +b ^(2) lt c ^(3)`
B. `a ^(3) +b ^(3) gt 16c^(3)`
C. `a ^(3) +b ^(3) = 16 c ^(3)`
D. `a ^(3) + b ^(3)=c ^(3)`

Answer

To solve the problem, we need to establish a relationship between the segments \( a \) and \( b \) of a focal chord of a parabola and the semi-latus rectum \( c \).

### Step-by-Step Solution:

1. **Understanding the Focal Chord**:
In a parabola, a focal chord is a line segment that passes through the focus and has its endpoints on the parabola. The segments \( a \) and \( b \) are the lengths of the segments of this focal chord.

2. **Latus Rectum of the Parabola**:
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry of the parabola and passing through the focus. The length of the latus rectum is given as \( 4c \).

3. **Relationship Between Segments and Latus Rectum**:
The semi-latus rectum \( c \) is the harmonic mean of the segments \( a \) and \( b \) of the focal chord. The relationship can be expressed as:
\[ \frac{1}{a} + \frac{1}{b} = \frac{2}{l} \]
where \( l \) is the length of the latus rectum.

4. **Substituting the Length of the Latus Rectum**:
Since \( l = 4c \), we can substitute this into our equation:
\[ \frac{1}{a} + \frac{1}{b} = \frac{2}{4c} = \frac{1}{2c} \]

5. **Finding a Relationship**:
Rearranging the equation gives:
\[ \frac{a + b}{ab} = \frac{1}{2c} \]
Multiplying both sides by \( 2abc \) leads to:
\[ 2(a + b)c = ab \]

6. **Cubing Both Sides**:
To find a more complex relationship, we can cube both sides:
\[ (a + b)^3 \geq 64c^3 \]
This follows from the inequality between the arithmetic mean and the geometric mean.

7. **Final Result**:
Thus, we find that:
\[ a^3 + b^3 \geq 16c^3 \]

### Conclusion:
The relationship established is:
\[ a^3 + b^3 \geq 16c^3 \]