The straight line $x+y=k$ will be a tangent to the curve $16x^2+9y^2=144,$ if $k=$

$16x^2+9y^2=144\mathrm{............}\left(1\right)$ 

$x+y=k\mathrm{................}\left(2\right)$

If $y=mx+c$ is tangent of equation  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Then $\begin{array}{}{c}^2=a^2{m}^2+b^2\end{array}$

$a^2=9,b^2=16,m=-1$

$\begin{array}{} \end{array}\Rightarrow k^2=9+16=25$

$k=\pm 5$