<p>Tangent is drawn at any point \((p, q)\) on the parabola \(y^2 = 4ax\). Tangents are drawn from any point on this tangent to the circle \(x^2 + y^2 = a^2\), such that the chords of contact pass through a fixed point \((r, s)\). Then \(p, q, r, s\) hold which of the given relation?</p>
Step-by-Step Solution
Key Concept: The chord of contact from any point on the tangent to the parabola passes through a fixed point; this fixed point must satisfy the pole-polar relationship with respect to the circle, and its coordinates relate to the parabola point through the tangent equation.
<p><strong>Step 1:</strong> The tangent at point (p,q) on parabola y² = 4ax is: qy = 2a(x + p)</p><p><strong>Step 2:</strong> Any point on this tangent can be written as (x₀,y₀) where qy₀ = 2a(x₀ + p). The chord of contact from (x₀,y₀) to circle x² + y² = a² is: xx₀ + yy₀ = a²</p><p><strong>Step 3:</strong> For this chord to pass through fixed point (r,s), we need: xr + ys = a² to be satisfied by all valid (x₀,y₀).</p><p><strong>Step 4:</strong> Since (x₀,y₀) lies on the tangent qy₀ = 2a(x₀ + p), we can express: xx₀ + yy₀ = a² must be consistent with the tangent equation for all points on it passing through (r,s).</p><p><strong>Step 5:</strong> Comparing xx₀ + yy₀ = a² with the tangent equation qy = 2a(x + p), by pole-polar relationship: r = p and s = q, giving us the fixed point (p,q).</p><p><strong>Step 6:</strong> However, the fixed point satisfies: r² + s² = a² (since the pole of a chord w.r.t. circle lies on the director circle relationship). Also, since (p,q) is on the parabola: q² = 4ap.</p><p>∴ Answer: <strong>p = r, q = s, and r² + s² = a² with q² = 4ap</strong> (or equivalently s² + r² = a²)</p>
Correct Answer: D