I assume the equation of the plane is [tex]9x+5y+z=5[/tex] and the cylinder has equation [tex]\dfrac{x^2}{16}+\dfrac{y^2}{81}=1[/tex].
I don't know what techniques are available to you, so I'll resort to (in my opinion) the most reliable: surface integration.
The surface of intersection [tex]S[/tex] is an ellipse in three dimensional space which can be parameterized by [tex]\mathbf x(u,v)=\langle 4u\cos v,9u\sin v,5-36u\cos v-45 u\sin v\rangle[/tex], with [tex]u\in[0,1][/tex] and [tex]v\in[0,2\pi][/tex].
The area is then given by the integral
[tex]\displaystyle\iint_S\mathrm dA=\int_0^1\int_0^{2\pi}\left\|\dfrac{\partial\mathbf x}{\partial u}\times\dfrac{\partial\mathbf x}{\partial v}\right\|\,\mathrm dv\,\mathrm du=36\sqrt{107}\pi[/tex]
