Find the equation of an ellipse if its center is S(2, 1) and the edges of a triangle PQR are tangent lines to this ellipse. P(0, 0), Q(5, 0), R(0, 4).

My attempt: Let take a point on the line PQ. For example (m,0). Then we have an equation of a tangent line for this point: $({a}_{11}m+{a}_{1})x+({a}_{12}m+{a}_{2})y+({a}_{1}m+a)=0$, where ${a}_{11}$ etc are coefficients of our ellipse: ${a}_{11}{x}^{2}+2{a}_{12}xy+{a}_{22}{y}^{2}+2{a}_{1}x+2{a}_{2}y+a=0$. Now if PQ: y = 0, then $({a}_{11}m+{a}_{1})=0$ , ${a}_{12}m+{a}_{2}=1$, ${a}_{1}m+a=0$ .I've tried this method for other 2 lines PR and RQ and I got 11 equations (including equations of a center)! Is there a better solution to this problem?