Question

Find the relation between $a$ and $b$ by simplifying the expression $(3a+2b{)}^2+(2a+3b{)}^2$.

Also, find the value of expression when $a^2+b^2=5$ and $ab=2$.

• A. $113$
• B. $13(a^2+b^2)+24ab$
• C. $24(a^2+b^2)+13ab$
• D. $146$

Answer 1

Answer: (A), (B)

$13(a^2+b^2)+24ab$

Given the expression
$(3a+2b{)}^2+(2a+3b{)}^2$

Let $(3a+2b)=x$ and $(2a+3b)=y$
$\therefore$ the expression becomes $x^2+y^2$

$\Rightarrow x^2+y^2=(x+y{)}^2-2xy$
$[\because (a+b{)}^2=a^2+b^2+2ab]$

Now, re-substituting value of $x$ and $y$ in the above expression we get,
$\Rightarrow x^2+y^2=\left[\right(3a+2b)+(2a+3b){]}^2-2(3a+2b\left)\right(2a+3b)$
$\Rightarrow x^2+y^2=[3a+2b+2a+3b{]}^2-2[6a^2+13ab+6b^2]$
$\Rightarrow x^2+y^2=\left[5\right(a+b){]}^2-2[6(a^2+b^2)+13ab]$
$\Rightarrow x^2+y^2=25(a^2+b^2+2ab)-12(a^2+b^2)-26ab$
$\Rightarrow x^2+y^2=25(a^2+b^2)+50ab-12(a^2+b^2)-26ab$

$\Rightarrow x^2+y^2=13(a^2+b^2)+24ab$

So, this is the required relation between $a$ and $b$.

​Now,Putting value of $a^2+b^2=5$ and $ab=2$ we get

$(3a+2b{)}^2+(2a+3b{)}^2=13(a^2+b^2)+24ab=13\times 5+24\times 2$

$\Rightarrow 13(a^2+b^2)+24ab=65+48=113.$​​​​​​