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 find the relationship between the segments of a focal chord \( a \) and \( b \) of a parabola, given that the latus rectum is \( 4c \).

### Step-by-Step Solution:

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

2. **Semi-Latus Rectum**:
The semi-latus rectum \( L \) is half of the latus rectum. Therefore, we have:
\[ L = \frac{4c}{2} = 2c \]

3. **Harmonic Mean Relation**:
The semi-latus rectum is the harmonic mean of the segments \( a \) and \( b \) of the focal chord. The formula for the harmonic mean \( HM \) of two numbers \( a \) and \( b \) is given by:
\[ HM = \frac{2ab}{a + b} \]
Setting this equal to the semi-latus rectum, we have:
\[ 2c = \frac{2ab}{a + b} \]

4. **Cross-Multiplying**:
Cross-multiplying the equation gives:
\[ 2c(a + b) = 2ab \]
Simplifying this, we find:
\[ c(a + b) = ab \]

5. **Rearranging the Equation**:
Rearranging the equation, we get:
\[ ab - ca - cb = 0 \]
This can be factored as:
\[ a(b - c) + b(c - a) = 0 \]

6. **Using AM-GM Inequality**:
We know from the properties of means that:
\[ AM \geq GM \geq HM \]
where \( AM = \frac{a + b}{2} \), \( GM = \sqrt{ab} \), and \( HM = \frac{2ab}{a + b} \).

7. **Applying the Inequality**:
Since \( HM = 2c \), we can write:
\[ \frac{a + b}{2} \geq 2c \]
Multiplying through by 2 gives:
\[ a + b \geq 4c \]

8. **Cubing the Sums**:
We can also relate \( a^3 + b^3 \) to \( c \). Using the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
and knowing \( a^2 + b^2 = (a + b)^2 - 2ab \), we can derive:
\[ a^3 + b^3 \geq 16c^3 \]

9. **Final Result**:
Therefore, we conclude that:
\[ a^3 + b^3 > 16c^3 \]

### Conclusion:
The relationship we derived is:
\[ a^3 + b^3 > 16c^3 \]