<p>The equation of the tangent at the point '\(\theta\)' on the hyperbola \(4x^2 - 3y^2 = 12\) is ___ where \(\theta = \pi/4\).</p>
Step-by-Step Solution
Key Concept: For hyperbola x²/a² - y²/b² = 1, the parametric point is (a·sec θ, b·tan θ), and the tangent equation is (x·sec θ)/a - (y·tan θ)/b = 1. Substitute the given θ value directly into this standard form.
<p><strong>Step 1:</strong> Convert to standard form.</p><p>4x² - 3y² = 12</p><p>x²/3 - y²/4 = 1</p><p>Here a² = 3, b² = 4, so a = √3, b = 2</p><p><strong>Step 2:</strong> Identify parametric point at θ = π/4.</p><p>Parametric form: x = a·sec θ = √3·sec(π/4) = √3·√2 = √6</p><p>y = b·tan θ = 2·tan(π/4) = 2·1 = 2</p><p><strong>Step 3:</strong> Apply tangent formula.</p><p>Tangent at parameter θ: (x·sec θ)/a - (y·tan θ)/b = 1</p><p>(x·sec(π/4))/√3 - (y·tan(π/4))/2 = 1</p><p>(x·√2)/√3 - (y·1)/2 = 1</p><p><strong>Step 4:</strong> Simplify.</p><p>(√2·x)/√3 - y/2 = 1</p><p>Multiply by 6: 2√6·x - 3y = 6</p><p>∴ <strong>Answer: 2√6·x - 3y = 6</strong> or equivalently <strong>(√2·x)/√3 - y/2 = 1</strong></p>
Correct Answer: 2