Closed-Form Solution of Conic in Point-Line Enumerative Problem of Conic

We use the reciprocal transformation to propose the closed-form solutions to the conics through m points and tangent to n lines satisfying m+n=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m+n=5$$\end{document} in general position. We also derive the algebraic and geometric necessary and sufficient conditions for the non-degenerate real conics.