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Expansion of (a ± b ± c)²

If a2+b2=13...

Question

If $a^{2} + b^{2} = 13$ and $a b = 6$ find $a^{2} - b^{2}$ .

A

\pm 9

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B

\pm 5

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C

\pm 2

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D

\pm 6

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Solution

The correct option is B $\pm 5$
$(a + b)^{2} = a^{2} + b^{2} + 2 a b$
$= 13 + 2 (6)$
$= 13 + 12$
$= 25$
$=> (a + b) = \pm \sqrt{25}$
$=> (a + b) = \pm 5$
Now,
$(a - b)^{2} = a^{2} + b^{2} - 2 a b$
$= 13 - 2 (6)$
$= 13 - 12$
$= 1$
$=> (a - b) = \pm 1$
Now,
$a^{2} - b^{2} = (a + b) (a - b)$
$= (\pm 1) (\pm 5)$
$= \pm 5$

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