= Solution
The projected quasar-to-cloud separation is
$$
R_\perp=10\left(\frac{\pi}{180\times3600}\right)D_A=87.6\ \mathrm{kpc}.
$$
For the usual transverse-separation estimate, take the cloud to lie at the same line-of-sight proper distance as the <quasar>. Since the cloud and galaxy are gravitationally bound, the local source-to-cloud path does not acquire the <redshift> factor $1+z=3$ that separates that system from us. Neglect internal relative Doppler shifts and attenuation. The observer in the cloud sees
$$
\boxed{\lambda_{\rm cloud}\simeq200\ \mathrm{nm},\qquad
F_{\nu,\rm cloud}\simeq\frac{L_{\nu,\rm em}}{4\pi R_\perp^2}=1.15\times10^{-20}\ \mathrm{W\,m^{-2}\,Hz^{-1}}}.
$$
This uses the <quasar>'s isotropic intrinsic emission at the same rest <frequency>; anisotropic radiation or obscuration would modify it.
An angular separation only fixes $R_\perp$. If $s$ is a source-to-cloud displacement toward us, the true local distance is $r=\sqrt{R_\perp^2+s^2}$ and the general flux is $L_\nu/(4\pi r^2)$. The quoted transverse value is therefore an upper estimate at fixed projected separation. Equality of cosmological <redshift> and gravitational binding do not by themselves measure the small internal line-of-sight displacement; a <peculiar velocity> can also mask it.
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