<p>Let \(P_0\) is the parabola \(y^2 = 4x\) with vertex \(K(0,0)\), \(A\) and \(B\) are points on \(P_0\) where tangents drawn intersect at right angles. Let \(C\) be the centroid of \(\triangle ABK\). The locus of \(C\) is another parabola \(P_1\). Now the process is repeated with \(P_1\) then \(P_2, P_3, \ldots\) etc. Then the length of latus rectum of \(P_{10}\) can be expressed as \(\dfrac{a}{b}\) where \(a, b\) are co-prime natural numbers. Find the value of \((a + \log_3 b)\).</p>
Step-by-Step Solution
Key Concept: When tangents to a parabola intersect at right angles, they meet on the directrix. By finding the locus of centroids of triangles formed by such points and the vertex, we get a new parabola with a specific scaling relationship. This scaling pattern repeats, creating a geometric sequence for the latus rectum.
<p><strong>Step 1: Find the condition for perpendicular tangents on P₀: y² = 4x</strong></p><p>For parabola y² = 4x, the tangent at point (t₁², 2t₁) is: y = x/t₁ + t₁</p><p>Similarly, tangent at (t₂², 2t₂) is: y = x/t₂ + t₂</p><p>For perpendicular tangents: (1/t₁)(1/t₂) = -1 ⟹ t₁t₂ = -1</p><p>The tangents intersect on the directrix x = -1.</p><p><strong>Step 2: Find coordinates of A and B</strong></p><p>Let A = (t₁², 2t₁) and B = (t₂², 2t₂) where t₂ = -1/t₁</p><p>So B = (1/t₁², -2/t₁), K = (0, 0)</p><p><strong>Step 3: Find centroid C of triangle ABK</strong></p><p>C = ((t₁² + 1/t₁² + 0)/3, (2t₁ - 2/t₁ + 0)/3)</p><p>Let x = (t₁² + 1/t₁²)/3 and y = (2t₁ - 2/t₁)/3</p><p>From the y-coordinate: 3y = 2(t₁ - 1/t₁) ⟹ t₁ - 1/t₁ = 3y/2</p><p>(t₁ - 1/t₁)² = 9y²/4 ⟹ t₁² + 1/t₁² - 2 = 9y²/4</p><p>t₁² + 1/t₁² = 2 + 9y²/4</p><p><strong>Step 4: Derive equation of P₁</strong></p><p>From x = (t₁² + 1/t₁²)/3: 3x = 2 + 9y²/4</p><p>3x - 2 = 9y²/4</p><p>y² = (4/9)(3x - 2) = (4/3)x - 8/9</p><p>This is P₁: y² = (4/3)x - 8/9, which can be rewritten with vertex shifted.</p><p><strong>Step 5: Identify the latus rectum pattern</strong></p><p>P₀: y² = 4x has latus rectum = 4</p><p>For P₁ in standard form (after shifting vertex): The parameter a₁ = 1/3, so latus rectum L₁ = 4/3</p><p>Pattern: Each iteration multiplies latus rectum by 1/4 (equivalently, parameter becomes 1/4 of previous)</p><p>L₀ = 4, L₁ = 4/3, L₂ = 4/9, ..., Lₙ = 4/3ⁿ</p><p><strong>Step 6: Calculate L₁₀</strong></p><p>L₁₀ = 4/3¹⁰</p><p>In lowest terms: a = 4, b = 3¹⁰ = 59049</p><p>gcd(4, 59049) = 1 (since 59049 is odd) ✓</p><p><strong>Step 7: Calculate final answer</strong></p><p>a + log₃(b) = 4 + log₃(3¹⁰) = 4 + 10 = 14</p><p>∴ Answer: 14</p>
Correct Answer: 14