<p>The curves \(x = a(1 + \cos\theta)\) and \(y = a\sin\theta\). The equation of the normal at \(\theta\) passes through the point</p>
Step-by-Step Solution
Key Concept: For a parametric curve, the normal line is perpendicular to the tangent. Find dy/dx using parametric derivatives, then use the perpendicular slope to write the normal equation and determine which fixed point it always passes through.
<p><strong>Step 1: Find the derivatives</strong></p><p>Given: x = a(1 + cos θ), y = a sin θ</p><p>dx/dθ = -a sin θ</p><p>dy/dθ = a cos θ</p><p><strong>Step 2: Find dy/dx</strong></p><p>dy/dx = (dy/dθ)/(dx/dθ) = (a cos θ)/(-a sin θ) = -cot θ</p><p><strong>Step 3: Find the slope of the normal</strong></p><p>Slope of normal = -1/(dy/dx) = tan θ</p><p><strong>Step 4: Write the equation of normal at point (a(1 + cos θ), a sin θ)</strong></p><p>y - a sin θ = tan θ · (x - a(1 + cos θ))</p><p>y - a sin θ = tan θ · x - a tan θ(1 + cos θ)</p><p>y = tan θ · x - a tan θ(1 + cos θ) + a sin θ</p><p><strong>Step 5: Simplify the constant term</strong></p><p>Constant = -a tan θ(1 + cos θ) + a sin θ</p><p>= -a(sin θ/cos θ)(1 + cos θ) + a sin θ</p><p>= a sin θ[-（1 + cos θ)/cos θ + 1]</p><p>= a sin θ[-(1 + cos θ) + cos θ]/cos θ</p><p>= a sin θ[-1]/cos θ = -a tan θ</p><p><strong>Step 6: Identify the fixed point</strong></p><p>The normal can be rewritten as: y = tan θ · x - a tan θ</p><p>y = tan θ(x - a)</p><p>This passes through the point <strong>(a, 0)</strong> for all values of θ.</p><p>∴ Answer: C (The point is (a, 0))</p>
Correct Answer: C