Intersections and algebraic tangency

Solve conic intersections and establish tangency using algebra rather than calculus. This precalculus set combines substitution, elimination, quadratic discriminants and geometric checks for circles, ellipses, hyperbolas and parabolas. Single-select and multiselect problems test all real intersection points, tangent-line parameters, vertical cases and coincident equations. Detailed explanations show when a repeated-root argument applies and why one intersection alone does not always establish tangency.