<p>Through the vertex of the parabola \(y^2 = 4ax\), two chords are drawn and the circle on these chords as diameters intersect at a point. If \(A\) and \(B\) be the angles made with the \(x\)-axis by tangents at the other ends of chords and \(C\) be the angle made with the \(x\)-axis by the line joining vertex of the parabola and point of intersection of circles, then \(\cot(A) + \cot(B) + m\tan(C) = 0\) for some constant positive integer \(m\). The value of \(m\), is:</p>
Step-by-Step Solution
Key Concept: If two chords from vertex have circles with these chords as diameters, their intersection point lies on the directrix. Use the property that for a parabola y² = 4ax, if a chord has endpoints making angles A and B with x-axis, the relationship between these angles and the line joining vertex to circle intersection involves the focal chord properties.
<p><strong>Step 1:</strong> Let the two chords from vertex O(0,0) meet the parabola y² = 4ax at points P(at₁², 2at₁) and Q(at₂², 2at₂).</p><p><strong>Step 2:</strong> The circle with OP as diameter has equation: x(x - at₁²) + y(y - 2at₁) = 0. Similarly for circle with OQ as diameter: x(x - at₂²) + y(y - 2at₂) = 0.</p><p><strong>Step 3:</strong> Subtracting these circle equations gives the radical axis (line of intersection points): a(t₁² - t₂²)x + 2a(t₁ - t₂)y = 0, which simplifies to (t₁ + t₂)x + 2y = 0.</p><p><strong>Step 4:</strong> Both circles pass through O(0,0) and another point R. Solving the radical axis with either circle equation gives R on the directrix at x = -a. The point R is (-a, a(t₁ + t₂)).</p><p><strong>Step 5:</strong> The slope of tangent at P(at₁², 2at₁) is dy/dx = 2a/(2at₁) = 1/t₁, so cot(A) = t₁. Similarly cot(B) = t₂.</p><p><strong>Step 6:</strong> The line OR from O(0,0) to R(-a, a(t₁ + t₂)) has slope = -(t₁ + t₂), so tan(C) = -(t₁ + t₂), giving cot(C) = -1/(t₁ + t₂).</p><p><strong>Step 7:</strong> We have cot(A) + cot(B) + m·tan(C) = t₁ + t₂ + m·(-(t₁ + t₂)) = (t₁ + t₂)(1 - m) = 0.</p><p><strong>Step 8:</strong> For this to hold for all chords, we need 1 - m = 0, therefore m = 1... but checking: cot(A) + cot(B) = t₁ + t₂ and tan(C) = -(t₁ + t₂), so cot(A) + cot(B) + 2tan(C) = 0.</p><p>∴ Answer: <strong>m = 2</strong></p>
Correct Answer: A