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JEE Previous Year Question

$If x + y = k, x > 0, y > 0 then x^{2} + y^{2} is minimum, if$

Topic: JEE PYQ

Answer: C. Full step-by-step solution below.

A $x > y$
B $x < y$
C $x = y$
D $x = 2 y$

Correct answer: C

Solution

(c) x + y = k \Rightarrow y = k - x \dots \dots .. (i) let S = x^{2} + y^{2} = x^{2} + (k - x)^{2} \frac{ds}{dx} = 2 x - 2 (k - x) = 4 x - 2 k = 0 \Rightarrow x = k /2 \frac{d ^{2} s}{d x ^{2}} = u > 0 ∴ S is minimum at x = \frac{k}{2} from (i) y = k - \frac{k}{2} = \frac{k}{2} = x

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