$\because$ Equation of ellipse is $9x^{2} +16y^{2} = 144$ or
$\frac{x^{2}}{16}+\frac{y^{2}}{9} = 1$
Comparing this with $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ then we
get $a^{2} = 16$ and $b^{2}=9$ and comparing the line
$y = x + \lambda$ with $y = mx + c$
$\therefore m = 1$ and $c = \lambda$
If the line $y = x + \lambda$ touches the ellipse $9x^{2} $
$+ 16y^{2} = 144$, then $c^{2} = a^{2}m^{2} + b^{2}$
$\Rightarrow \quad\lambda^{2}= 16 \times 1^{2} + 9 \Rightarrow \lambda^{2} = 25$
$\therefore \quad\lambda = \pm 5$