Test whether a tangent meets the curve again

$dx/dt=3$, $dy/dt=1-3/t^2$, so $dy/dx=1/3-1/t^2$. At $x=11$, $t=3$, $P=(11,4)$ and the gradient is $2/9$. The tangent is $y-4=(2/9)(x-11)$, or $y=2x/9+14/9$.

Substitute the parametric equations into the tangent: $t+3/t=2(3t+2)/9+14/9$. Since $t\ne0$, clearing denominators gives $3(t^2+3)=t(2t+6)$, so $t^2-6t+9=(t-3)^2=0$. Only $t=3$ occurs, and $x=3t+2$ is one-to-one in $t$. Thus the tangent meets the curve only at $P$ and does not cut it again.