The line $5x-2y+4k=0$ is tangent to $4x^{2}-y^{2}=36$, then k is:

The equation of the hyperbola is

$4x^2-y^2=36$

$\dfrac{4x^2}{36}-\dfrac{y^2}{36}=1$

$\dfrac{4x^2}{9}-\dfrac{y^2}{36}=1$............(1)

$a^2=9;$

$b^2=36$

The equation of the line is

$5x-2y+4k=0$

$2y=5x+4k$

$y=\dfrac{5}{2}x+2k$............(2)

From equation (2) $m=\dfrac{5}{2},\,c=2k$

$c^2=a^2m^2-b^2$

$(2k)^2=9(\dfrac{5}{2})^2-36$

$4k^2=9\times \dfrac{25}{4}-36$

$4k^2=\dfrac{225-144}{4}$

$k^2=\dfrac{81}{16}\Rightarrow k=\dfrac{9}{4}$