Parabola

The graphical method makes it possible to interpret an equation or an inequality through the graph of one or more functions, turning an algebraic problem into a geometric one. For quadratic equations, this interpretation is particularly direct: the solutions correspond to the intersections between a parabola and the x-axis, or to the intersections between two … Read more

Finding the tangent line to a parabola at a given point is one of the core topics of analytic geometry, and it is an essential step toward understanding the conic sections. In particular, this article shows how to determine the equation of a line that is tangent to a given parabola and that also satisfies … Read more

This article studies the relative position of a line and a parabola. In the Cartesian plane, a line and a parabola can occupy different relative positions: the line can be tangent, secant, or external to the parabola. The goal of this article is to provide an analytic approach to the problem for both the parabola … Read more

The study of the parabola in analytic geometry occasionally leads to a special case: the parabola whose axis of symmetry is parallel to the x-axis, also known as a parabola with a horizontal axis or, more simply, a horizontal parabola. This article first introduces the theory behind the parabola with a horizontal axis, starting from … Read more

Finding the equation of a parabola that passes through three given points is one of the classic, fundamental problems of analytic geometry. In this article, in particular, we will see how to find the equation of a parabola with a vertical axis passing through three points, using the standard algebraic method: solving a linear system … Read more