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Harmonic Progression

If x+y=16 and...

Question

If x+y =16 and $x^{2} + y^{2}$ is minimum, then the values of x and y are

3, 13

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4, 12

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6, 10

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8, 8

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Solution

The correct option is D 8, 8
$x + y = 16 \Rightarrow y = 16 - x \Rightarrow x^{2} + y^{2} = x^{2} + (16 - x)^{2}$
Let $z = x^{2} + (16 - x)^{2} \Rightarrow z^{'} = 4 x - 32$
To be minimum of z, z' = 0, .
Therefore $4 x - 32 = 0 \Rightarrow x = 8 \Rightarrow y = 8$

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