I assume the equation of the plane is \(9x+5y+z=5\) and the cylinder has equation \(\dfrac{x^2}{16}+\dfrac{y^2}{81}=1\).
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 \(S\) is an ellipse in three dimensional space which can be parameterized by \(\mathbf x(u,v)=\langle 4u\cos v,9u\sin v,5-36u\cos v-45 u\sin v\rangle\), with \(u\in[0,1]\) and \(v\in[0,2\pi]\).
The area is then given by the integral
\(\displaystyle\iint_S\mathrm dA=\int_0^1\int_0^{2\pi}\left\|\dfrac{\partial\mathbf x}{\partial u}\times\dfrac{\partial\mathbf x}{\partial v}\right\|\,\mathrm dv\,\mathrm du=36\sqrt{107}\pi\)
