JEEOlympiad
Let x and y be positive real numbers with x^2 + y^2 = 1. Find the…
Question
Let x and y be positive real numbers with x^2 + y^2 = 1. Find the minimum value of 4/x^2 + 9/y^2.
✓ Verified answer: 25checked by our engine — not a guess
Step-by-step solution
By the Cauchy-Schwarz inequality, (4/x^2 + 9/y^2)(x^2 + y^2) >= (2 + 3)^2 = 25.
Since x^2 + y^2 = 1, we get 4/x^2 + 9/y^2 >= 25.
Equality holds when (4/x^2)/x^2 = (9/y^2)/y^2, i.e.
2/x^2 = 3/y^2, giving x^2 = 2/5, y^2 = 3/5 (which satisfy x^2 + y^2 = 1).
Hence the minimum value is 25.
Final answer25
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