<p>If \(n_1\) is the number of points on the line \(3x + 4y = 5\) which is at distance of \(1 + \sin^2\theta\) units from (2, 3) and \(n_2\) denotes the number of points on the line \(3x + 4y = 5\) which is at distance of \(\sec^2\theta + 2\csc^2\theta\) units from (1, 3), then find the sum of roots of equations \(n_2x^2 - 6x + n_1 = 0\).</p>
Step-by-Step Solution
Key Concept: We need to find how many points on a given line can be at a specific distance from a point. This requires analyzing the range of the distance expressions and comparing with the minimum distance from the point to the line. The number of intersection points depends on whether the distance value falls within the achievable range.
<p><strong>Step 1: Find perpendicular distance from (2,3) to line 3x + 4y = 5</strong></p><p>Distance = |3(2) + 4(3) - 5|/√(9+16) = |6 + 12 - 5|/5 = 13/5 = 2.6 units</p><p><strong>Step 2: Analyze distance d₁ = 1 + sin²θ</strong></p><p>Since 0 ≤ sin²θ ≤ 1, we have 1 ≤ 1 + sin²θ ≤ 2</p><p>The range [1, 2] is entirely less than 2.6 (the perpendicular distance)</p><p>Therefore, no circle with radius in [1,2] intersects the line</p><p>∴ n₁ = 0</p><p><strong>Step 3: Find perpendicular distance from (1,3) to line 3x + 4y = 5</strong></p><p>Distance = |3(1) + 4(3) - 5|/5 = |3 + 12 - 5|/5 = 10/5 = 2 units</p><p><strong>Step 4: Analyze distance d₂ = sec²θ + 2csc²θ</strong></p><p>Using AM-GM or calculus: Let f(θ) = sec²θ + 2csc²θ = 1/cos²θ + 2/sin²θ</p><p>Taking derivative and setting to 0, or using substitution u = cos²θ:</p><p>f(θ) = 1/u + 2/(1-u) where 0 < u < 1</p><p>Minimum occurs when d/du[1/u + 2/(1-u)] = 0</p><p>-1/u² + 2/(1-u)² = 0 ⟹ (1-u)² = 2u² ⟹ 1-u = √2·u ⟹ u = 1/(1+√2)</p><p>Minimum value = (1+√2) + 2√2 = 1 + 3√2 ≈ 5.24</p><p>As θ → 0 or π/2, the expression → ∞</p><p>So d₂ ∈ [1 + 3√2, ∞) where 1 + 3√2 ≈ 5.24</p><p><strong>Step 5: Compare d₂ range with perpendicular distance 2</strong></p><p>Since the minimum of d₂ is approximately 5.24 > 2, all values of d₂ exceed the perpendicular distance</p><p>Therefore, every circle with radius d₂ intersects the line at exactly 2 points</p><p>∴ n₂ = 2</p><p><strong>Step 6: Find sum of roots of n₂x² - 6x + n₁ = 0</strong></p><p>The equation becomes: 2x² - 6x + 0 = 0</p><p>By Vieta's formulas, sum of roots = 6/2 = 3</p><p><strong>∴ Answer: 3</strong></p>
Correct Answer: 3