If $a^{2} - b^{2} = 21$ and $a^{2} + b^{2} = 29$ which of the following could be the value of ab?
I. -10 II. $5 \sqrt{2}$ III.10

I only

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II only

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III only

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I and III only

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Solution

The correct option is D I and III only
$a^{2} + b^{2} = 29$ ...(i)
$a^{2} - b^{2} = 21$ ...(ii)
on adding (i) and (ii), we get
$2 a^{2} = 50$
$\Rightarrow a^{2} = 25$
$\Rightarrow a = \pm 5$
from(i), $b^{2} = 25 - 21 = 4$
$\Rightarrow b = \pm 2$
Clearly, when $a = 5, b = 2,$ then $a b = 10$
when $a = - 5, b = 2,$ then $a b = - 10$
Option D is correct.

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