Question

The descending order of magnitude of the eccentricities of the following hyperbolas is: 
A. A hyperbola whose distance between foci is three times the distance between its directrices. 
B. Hyperbola in which the transverse axis is twice the conjugate axis. 
C. Hyperbola with asymptotes \( x + y + 1 = 0, x - y + 3 = 0 \).

• A. \( C, A, B \)
• B. \( B, C, A \)
• C. \( \) (No option provided)
• D. \( A, C, B \)

Hint:

The eccentricity of a hyperbola is always greater than 1 and can be determined using the transverse and conjugate axis relations.

Answer 1

The Correct Option is D

Step 1: Finding the eccentricities For a hyperbola, the eccentricity is given by: \[ e = \frac{\text{distance between foci}}{\text{length of transverse axis}}. \] Solving for each case: 
- Hyperbola A: Given condition leads to \( e = \frac{3}{2} \). 
- Hyperbola B: \( e = \frac{\sqrt{5}}{2} \). 
- Hyperbola C: Given asymptotes suggest \( e = \sqrt{2} \). 
Ordering the values, we get: \[ A > C > B. \]