Equation of hyperbola is $\large\frac{x^2}{9}-\frac{y^2}{16}$$=1$
Hence $a^2=9,b^2=16$
Comparing the line $y=2x+\lambda$ with $y=mx+c$
$m=2,c=\lambda$
If the line $y=2x+\lambda$ touches the hyperbola $\large\frac{x^2}{9}-\frac{y^2}{16}$$=1$ if $c^2=a^2m^2-b^2$
$\lambda^2=9(2)^2-16=20$
$\lambda=\pm 2\sqrt 5$
Hence (a) is the correct answer.