Math

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

Determining the relative position of a line and a circle is an important problem in analytic geometry. It is fairly intuitive that the relative position of a line and a circle depends essentially on the number of points they have in common in the Cartesian plane. In particular, three fundamental cases arise: For the full … Read more

Finding the equation of a circle through three points is one of the classic problems in analytic geometry. In particular, given three distinct points in the Cartesian plane that are not collinear, it is possible to determine the equation of the circle that contains all three (that is, the circle passing through all of them) … Read more

The circle is one of the most widely studied figures in analytic geometry, both for its theoretical importance and for its role in countless exercises. This guide covers how to derive the equation of a circle, how to find its center and radius, and how to determine whether a given equation actually represents a circle. … Read more