# For what value of $\lambda$ does the line $y=2x+\lambda$ touches the hyperbola $16x^2-9y^2=144$?

$\begin{array}{1 1}(a)\;\pm 2\sqrt 5\\(b)\;\pm 2\sqrt 7\\(c)\;\pm 3\sqrt 5\\(d)\;\pm 4\sqrt 5\end{array}$

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.