Math Explained Clearly
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
In a previous article, we examined the parabola with vertex at the origin, whose axis coincides with the y-axis. This article represents the natural extension — and generalization — of that discussion. We will remove the assumption that the vertex \(V\) coincides with the origin \(O\) of the coordinate plane, and turn our attention to … Read more
The equation of a parabola with vertex at the origin is one of the foundational topics in analytic geometry. Knowing how to identify the vertex, axis of symmetry, direction of opening, focus, and directrix of a parabola makes it possible to fully analyze this important conic section and to correctly interpret its graph on the … Read more
The circle is one of the most extensively studied figures in plane coordinate geometry. It belongs to the family of conic sections (which also includes the parabola, ellipse, and hyperbola) and appears throughout coordinate geometry, from tangency problems to intersections between lines and curves. Studying the circle makes it possible to address a wide range … Read more
The circle is one of the most important topics in analytic geometry. This collection offers circle practice problems with step-by-step solutions, each worked through in full detail with a clear explanation of every stage. The problems cover the main question types: the equation of a circle, center and radius, whether a point lies on the … Read more
In the study of analytic geometry, the circle is not limited to representing a geometric locus — it also becomes an interpretive tool for tackling problem types that, on the surface, belong to algebraic analysis. Among these problems, radical inequalities occupy a place of particular importance. When a radical inequality contains an expression of the … Read more
A pencil of circles, sometimes also called a family of circles, is a powerful and elegant tool in analytic geometry. It provides a way to describe infinitely many circles with a single equation and to solve, in an elegant way, many problems, such as determining a circle through given points or studying particular geometric configurations. … Read more
Studying the relative position of two circles means determining how they are arranged in the Cartesian plane, in particular whether they intersect at two points, are tangent at one point, or are completely separate (with no points of intersection). This problem plays a central role in analytic geometry and can be approached from two complementary … Read more
Finding the equation of a line tangent to a circle is one of the most common — and most important — problems in analytic geometry, and it represents a fundamental step in the study of the relationship between lines and curves. Many exercises ask for the equation of a line tangent to a circle, either … Read more