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 \( x + y + 1 = 0, x - y + 3 = 0 \).
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 \( x + y + 1 = 0, x - y + 3 = 0 \).
Show Hint
The eccentricity of a hyperbola is always greater than 1 and can be determined using the transverse and conjugate axis relations.
Updated On: Apr 25, 2025
- \( C, A, B \)
- \( B, C, A \)
- \( \) (No option provided)
- \( A, C, B \)
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The Correct Option is D
Solution and Explanation
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: \[ A > C > B. \]
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