The length of transverse and conjugate axis in a hyperbola are 6 and 8, respectively. The eccentricity of the hyperbola, rounded off to TWO decimal places, is ________.
Key Formulas
For a hyperbola x2/a2 – y2/b2 = 1, the transverse axis length equals 2a and conjugate axis length equals 2b.
Eccentricity e satisfies e = √(1 + b2/a2) where e > 1. These relations hold for the standard form along the x-axis; the vertical orientation swaps roles but preserves the formula.
Step-by-Step Solution
Given 2a = 6, compute a = 3. Given 2b = 8, compute b = 4.
Substitute into the eccentricity formula: e = √(1 + 42/32) = √(1 + 16/9) = √(25/9) = 5/3 ≈ 1.6667. Rounded to two decimal places, e = 1.67.
No options are provided in the query, so none require evaluation; the computation confirms the numerical answer directly.
Verification
The result exceeds 1, consistent with hyperbola properties since b > a here implies e > √2.
Direct computation via code yields identical output, ensuring precision. Alternative verification: e2 = c2/a2 where c = √(a2 + b2) = √(9 + 16) = 5, so e = 5/3 confirms.
Understanding Hyperbola Axes
A hyperbola features two axes: the transverse axis (length 2a, along which vertices lie) and conjugate axis (length 2b, perpendicular).
For x2/a2 – y2/b2 = 1, 2a = 6 gives a = 3; 2b = 8 gives b = 4. These parameters define the curve’s shape and eccentricity.
Eccentricity Formula Explained
Hyperbola eccentricity e measures deviation from a circle (e > 1). Use e = √(1 + b2/a2).
Here: b2/a2 = 16/9, so e = √(1 + 1.7778) = √2.7778 = 1.6667 ≈ 1.67. Foci lie at ((±ae, 0)), with ae = 5.
Detailed Calculation Steps
- Identify: 2a = transverse axis = 6 → a = 3.
- Identify: 2b = conjugate axis = 8 → b = 4.
- Compute ratio: b2/a2 = 16/9 = 1.78.
- Add 1: 1 + 1.78 = 2.78.
- Square root: √2.78 = 1.67 (rounded).
Why This Matters for Exams
CSIR NET and similar tests emphasize hyperbola eccentricity computations. Common pitfalls: confusing 2a/2b or rounding errors.
Verification: e2 = c2/a2 where c = 5, so e = 5/3 confirms.
Related Properties
- Latus rectum: 2b2/a = 32/3 ≈ 10.67.
- Directrices: x = ±a/e = ±1.8.
- Practice variations like rectangular hyperbolas (a = b, e = √2) to master.
