Question

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$, then find the value of a and b.

• A. $a=11,b=-6$
• B. $a=10,b=-2$
• C. $a=11,b=-4$
• D. $a=12,b=-3$

Answer 1

The correct option is A $a=11,b=-6$

L.S.H,
$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$
The conjugate of $7+4\sqrt{3}$ is $7-4\sqrt{3}$
$=\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\times$
$\frac{7-4\sqrt{3}}{7-4\sqrt{3}}$
$=\frac{35-20\sqrt{3}+14\sqrt{3}-24}{49-(16\times 3)}$
$=\frac{11-6\sqrt{3}}{1}$
Equating to R.H.S
$=a+b\sqrt{3}$
$a=11,b=-6$