Question:

<p><strong>Step 1: Find dy/dx</strong></p><p>Given: x = a cos³θ, y = a sin³θ</p><p>dx/dθ = -3a cos²θ sin θ</p><p>dy/dθ = 3a sin²θ cos θ</p><p>dy/dx = (3a sin²θ cos θ)/(-3a cos²θ sin θ) = -tan θ</p><p><strong>Step 2: Equation of tangent at parameter θ</strong></p><p>Point on curve: (a cos³θ, a sin³θ)</p><p>Tangent equation: y - a sin³θ = -tan θ(x - a cos³θ)</p><p>Simplifying: y = -x tan θ + a sin³θ + a cos³θ tan θ</p><p><strong>Step 3: Find intercepts P and Q</strong></p><p>x-intercept (P): Set y = 0</p><p>0 = -x tan θ + a sin³θ + a cos³θ tan θ</p><p>x = a sin³θ/tan θ + a cos³θ = a sin²θ cos θ + a cos³θ = a cos θ(sin²θ + cos²θ) = a cos θ</p><p>So P = (a cos θ, 0)</p><p>y-intercept (Q): Set x = 0</p><p>y = a sin³θ + a cos³θ tan θ = a sin³θ + a cos²θ sin θ = a sin θ(sin²θ + cos²θ) = a sin θ</p><p>So Q = (0, a sin θ)</p><p><strong>Step 4: Find midpoint of PQ</strong></p><p>Midpoint M = ((a cos θ)/2, (a sin θ)/2)</p><p>Let (h, k) = midpoint</p><p>h = (a cos θ)/2 ⟹ cos θ = 2h/a</p><p>k = (a sin θ)/2 ⟹ sin θ = 2k/a</p><p><strong>Step 5: Eliminate θ</strong></p><p>cos²θ + sin²θ = 1</p><p>(2h/a)² + (2k/a)² = 1</p><p>4h²/a² + 4k²/a² = 1</p><p>4(h² + k²) = a²</p><p>∴ The locus is: <strong>x² + y² = a²/4</strong> (circle with radius a/2)</p>