From Circles to General Conics: Why the Tangent Substitution Works
The tangent equations for circles, ellipses, and hyperbolas share a noticeable pattern: replace one factor in each squared term with the corresponding coordinate of the point of contact, and the quadratic equation becomes a tangent line. Why does this work? How should we handle the linear term in a parabola, or the mixed term in a general quadratic curve? I want to start with a circle and work out the rule step by step. ...