Hyperbola
Eccentricity of Hyperbola
Grade 11
Question:
<p>A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the \(x\)-axis. Then the eccentricity of the hyperbola is __________ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Use the standard form of hyperbola x²/a² - y²/b² = 1, where transverse axis length = 2a gives us a directly, then substitute the given point to find b² and finally calculate eccentricity e = √(1 + b²/a²).
<p><strong>Step 1:</strong> Identify a from transverse axis length. Since transverse axis length = 2a = 4, we get <strong>a = 2</strong>.</p><p><strong>Step 2:</strong> Write the standard form: x²/4 - y²/b² = 1</p><p><strong>Step 3:</strong> Substitute point (4, 2) into the equation: 16/4 - 4/b² = 1</p><p>⟹ 4 - 4/b² = 1</p><p>⟹ 4/b² = 3</p><p>⟹ b² = 4/3</p><p><strong>Step 4:</strong> Use the hyperbola eccentricity formula: e² = 1 + b²/a² = 1 + (4/3)/4 = 1 + 1/3 = 4/3</p><p>⟹ e = √(4/3) = 2/√3 = 2√3/3</p><p><strong>Step 5:</strong> Calculate numerically: e = 2/√3 ≈ 1.1547</p><p>∴ Answer: <strong>1.1547</strong></p>
Correct Answer: 1