A geometry problem by Ayush G Rai

$\text{We shall use Secant-Tangent Rule, and Tangent of difference between angle formula.}\\ Note\ that\ \angle s\ in\ semicirces\ are\ 90^o. So \angle AFD=\angle BFA=90^o.\\ \implies\ D,\ F,\ B \ \ are\ colinear. \ \therefore\ F\ is \ on\ diagonal \ DB\ of \ 8\times 4\ rectangle\ ABCD.\\$ $\therefore\ DB=\sqrt{8^2+4^2}=4\sqrt5.........and..DP=4..........(1)\\ DP\ is\ tangent, \ DB\ and\ DF\ secants\ of green \ circle\ BFA.\\ \therefore\ DP^2=DF*DB, \implies\ form (1)\ DF=\dfrac{4^2} {4\sqrt5}= \dfrac4 {\sqrt5}...........(2)\\$ $\angle DCA= \angle PDB=say\ \alpha.\\ Sin(\alpha)=\frac {CB}{DB}=\frac 4 {4\sqrt5}=\frac 1 {\sqrt5},\ \ \implies\ Cos(\alpha)=\frac 2 {\sqrt5},\ \ Tan(\alpha)=\frac 1 2......(3)\\ \therefore\ EF=DF*Sin(\alpha)=\frac 4 5.......DE=DF*Cos(\alpha)=\frac 8 5.\ \ \ \ \implies\ EC=DC - DE=8 - \frac 8 5.\\$ $\therefore\ TanEDF=\dfrac{EF}{EC}=\frac 1 8.\\ \therefore\ Tan\theta=Tan(\alpha-EDF)=\dfrac{\frac 1 2 - \frac 1 8}{1+\frac 1 2 *\frac 1 8}=\dfrac 6 {17}=0.353.$

ans is 6/17 at first i did'nt saw that the 2nd line was not passing through origin! my mind completes the ques without seeing it wholly!that's why 1 failed attempt !