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MHT CET202520 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

If the curves y^2=6 x and 9 x^2+ by ^2=16 intersect each other at right angles, then the value of b is

Options

  1. A4
  2. B7 2
  3. C6
  4. D9 2

Correct answer

D. 9 2

Step-by-step solution

The curves C₁: y^2 = 6x and C₂: 9x^2 + by^2 = 16 intersect orthogonally when the product of their tangent slopes at the intersection point (x₀, y₀) equals -1 . For C₁ , 2y dy dx = 6 so the slope is dy dx ₁ = 3 y₀ . For C₂ , 18x + 2by dy dx = 0 so the slope is dy dx ₂ = -9x₀ by₀ . The orthogonality condition gives ( 3 y₀ )( -9x₀ by₀ ) = -1 , which simplifies to 27x₀ = by₀^2 . Since (x₀, y₀) lies on C₁ , we have y₀^2 = 6x₀ . Substituting yields 27x₀ = b(6x₀) . If x₀ = 0 , then C₂ gives 0 = 16 , a contradiction. Thus

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