The difference between the length 2a of the trans­verse axis of a hyperbola of eccentricity e and the length of its latus rectum is

Options

A. `2a|3-e^(2)|`
B. `2a|2-e^(2)|`
C. `2a|e^(2)-1|`
D. `a|2e^(2)-1|`

Answer

To find the difference between the length \(2a\) of the transverse axis of a hyperbola and the length of its latus rectum, we can follow these steps:

### Step 1: Identify the lengths involved
The length of the transverse axis of a hyperbola is given by:
\[ \text{Length of transverse axis} = 2a \]

The length of the latus rectum of a hyperbola is given by:
\[ \text{Length of latus rectum} = \frac{2b^2}{a} \]

### Step 2: Set up the difference
We need to find the difference between the length of the transverse axis and the length of the latus rectum:
\[ \text{Difference} = 2a - \frac{2b^2}{a} \]

### Step 3: Combine the terms
To combine these terms, we can express \(2a\) with a common denominator:
\[ \text{Difference} = \frac{2a^2}{a} - \frac{2b^2}{a} = \frac{2a^2 - 2b^2}{a} \]

### Step 4: Factor out the common terms
Factoring out the common factor of \(2\):
\[ \text{Difference} = \frac{2(a^2 - b^2)}{a} \]

### Step 5: Use the relationship between \(a\), \(b\), and eccentricity \(e\)
For a hyperbola, the relationship between the eccentricity \(e\), \(a\), and \(b\) is given by:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
From this, we can express \(b^2\) in terms of \(a\) and \(e\):
\[ b^2 = a^2(e^2 - 1) \]

### Step 6: Substitute \(b^2\) into the difference
Now, substitute \(b^2\) into the difference:
\[ \text{Difference} = \frac{2(a^2 - a^2(e^2 - 1))}{a} \]
This simplifies to:
\[ \text{Difference} = \frac{2a^2(1 - (e^2 - 1))}{a} = \frac{2a^2(2 - e^2)}{a} \]

### Step 7: Simplify the expression
Finally, we simplify:
\[ \text{Difference} = 2a(2 - e^2) \]

### Final Result
Thus, the difference between the length \(2a\) of the transverse axis and the length of its latus rectum is:
\[ \text{Difference} = 2a(2 - e^2) \]

---