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Maths · Trigonometrical identities and trigonometrical functions

Maximum value of where where are constants is

Maximum value of \( r \) where \( \frac{c^{2}}{r^{2}}= \) \( \frac{\mathbf{a}^{2}}{\sin ^{2} \theta}+\frac{b^{2}}{\cos ^{2} \theta} \) where \( c, a, b \) are constants is

  • A. \( \frac{c}{a+b} \)
  • B. \( \frac{c}{a-b} \)
  • C. \( \frac{a^{2}+b^{2}}{c^{2}} \)
  • D. \( \frac{c^{2}}{a^{2}+b^{2}} \)

Step-by-step solution

Given equation: c²/r² = a²/sin²θ + b²/cos²θ. To maximize r, minimize f(θ)=a²/sin²θ+b²/cos²θ. Using identity: a²/sin²θ+b²/cos²θ = a²(1+cot²θ)+b²(1+tan²θ) = a²+b² + a²cot²θ+b²tan²θ. By AM-GM, a²cot²θ+b²tan²θ ≥ 2ab, equality when tan²θ = a/b. Minimum f = (a+b)². Thus r_max = c/√((a+b)²) = c/(a+b).
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