Question

If $( a + b ) \cdot( a - b )=8$ and $| a |=8| b |$, then

• A. $| a |=\frac{16}{3} \sqrt{\frac{2}{7}}$
• B. $| b |=\frac{2}{3} \sqrt{\frac{2}{7}}$
• C. Both(a) and (b) are true
• D. Both(a) and (b) are false

Answer 1

Answer: (C)

Given, $(a+b) \cdot(a-b)=8$ and $|a|=8|b|$
$\Rightarrow a \cdot a-a \cdot b+b \cdot a-b \cdot b=8$
$\Rightarrow |a|^2-|b|^2=8 \left(\because a \cdot a=|a|^2\right.$ anda $\left.\cdot b=b \cdot a\right)$
$\Rightarrow (8| b |)^2-| b |^2=8$
$\Rightarrow 63|b|^2=8 ($ given, $|a|=8|b|)$
$\Rightarrow |b|=\sqrt{\frac{8}{63}}=\frac{2}{3} \sqrt{\frac{2}{7}}$
Also, $|a|=8|b|=8\left(\frac{2}{3} \sqrt{\frac{2}{7}}\right)=\frac{16}{3} \sqrt{\frac{2}{7}} .$