<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 a point on the tangent to the parabola passes through a fixed point if that point lies on the directrix of the parabola. Use the property that the chord of contact from point (h,k) to circle x²+y²=a² is hx+ky=a², and the tangent at (p,q) on parabola y²=4ax is qy=2a(x+p).
<p><strong>Step 1:</strong> Tangent at point (p,q) on parabola y²=4ax is: qy = 2a(x+p), where q²=4ap</p><p><strong>Step 2:</strong> Any point on this tangent can be written as (p+t²/4a, t) for parameter t. A general point on the tangent is (h,k) where qy=2a(x+p).</p><p><strong>Step 3:</strong> Chord of contact from point (h,k) to circle x²+y²=a² is: hx+ky=a²</p><p><strong>Step 4:</strong> For this chord to pass through fixed point (r,s) for ALL points (h,k) on the tangent qy=2a(x+p), the point (r,s) must satisfy the tangent equation itself.</p><p><strong>Step 5:</strong> Substituting (r,s) in tangent equation: qs=2a(r+p). Since q²=4ap, we get: qs=2ar+2ap, so qs=2ar+q²/2, giving s=2ar/q+q/2</p><p><strong>Step 6:</strong> For the chord of contact to always pass through (r,s), point (r,s) must lie on the directrix. The directrix of y²=4ax is x=-a. Therefore r=-a, and s can be any value.</p><p><strong>Step 7:</strong> The relation is: <strong>r=-a and s is arbitrary, or more specifically: qs=2a(r+p) with q²=4ap and r=-a</strong></p><p>∴ Answer: D</p>
Correct Answer: D