<p>If \(x\) and \(y\) are given parametrically, then \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{\sin t}{\cos t}\). The length of the tangent is:</p>
Step-by-Step Solution
Key Concept: The length of the tangent from a point on a curve to the x-axis is given by y/|dy/dx|, which simplifies using the parametric derivatives formula. For parametric curves, substitute the derivatives ratio directly into this formula.
<p><strong>Step 1:</strong> Recall that the length of the tangent from a point (x, y) on a curve to the x-axis is given by:</p><p>L = y/|dy/dx| or equivalently L = y·|dx/dy|</p><p><strong>Step 2:</strong> From the given information: dy/dx = sin t/cos t</p><p><strong>Step 3:</strong> Therefore: dx/dy = cos t/sin t</p><p><strong>Step 4:</strong> The length of tangent is:</p><p>L = y · |dx/dy| = y · |cos t/sin t|</p><p><strong>Step 5:</strong> Alternatively, using L = y/|dy/dx|:</p><p>L = y/|sin t/cos t| = y · |cos t/sin t|</p><p><strong>Step 6:</strong> This expression represents the length of tangent in parametric form, which equals y·cot t (when sin t > 0).</p><p>∴ Answer: A</p>
Correct Answer: A