Minimum possible value of $(x y)^{2} + (x + 7)^{2} + (2 y + 7)^{2},$ for all real $x$ and $y,$ is equal to

40

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47

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43

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45

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Solution

The correct option is D $45$
$x^{2} y^{2} + (x + 7)^{2} + (2 y + 7)^{2}$
$= (x^{2} + 4) (1 + y^{2}) + 14 (x + 2 y) + 94$
By Cauchy-Schwarz inequality,
$(x^{2} + 4) (1 + y^{2}) + 14 (x + 2 y) + 94$
$\geq (x + 2 y)^{2} + 14 (x + 2 y) + 94$
$= (x + 2 y + 7)^{2} + 45 \geq 45$
$∴$ Minimum value $= 45$

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