7. An ellipse has its major and minor axes parallel to the coordinate axes, and it is tangent to the line $2x + y = 11$ at point $P(4,3)$. It also passes through points $Q(0,-1)$ and $R(1, \sqrt{10} + 1)$. Find the equation of this ellipse.
7. Write $P(4,3)$ as the "point ellipse" equation $(x-4)^{2}+n(y-3)^{2}=0 \quad(n>0, n \neq 1)$, so the required reduced circle equation can be modified to $\lambda\left[(x-4)^{2}+n(y-3)^{2}\right]+u(2 x+y-11)=0$.

Take $\lambda=1$, and substitute the coordinates of points $Q$ and $R$ into the above equation to get $n=\frac{1}{2}, u=2$. Therefore, the required equation is $2(x-2)^{2}+(y-1)^{2}=12$.