Name: circle equations in standard form
Uploaded: 2025-11-26T05:44:19.000Z
Duration: 1 min 18 s
Channel: JoVE

Para determinar el radio del anillo de puntuación más grande de una diana, se puede modelar como un círculo.Un círculo consta de todos los puntos que tienen la misma distancia fija, llamada radio, desde un punto central conocido como centro.La ecuación de un círculo proviene de la fórmula de la distancia, que se basa en el teorema de Pitágoras. Mide la distancia entre el centro y cualquier punto del círculo. A medida que el punto P se mueve a lo largo del borde, su distancia desde el centro permanece constante. Al elevar al cuadrado ambos lados, se elimina la raíz cuadrada y se produce la forma estándar de la ecuación.Con el centro en el origen, la ecuación se reduce a la suma de los cuadrados de las coordenadas x e y.Para graficar un círculo a partir de su ecuación, las coordenadas del centro se seleccionan de la forma estándar, luego se identifica el radio.Por el contrario, dada una gráfica de un círculo, ubicar el centro y medir el radio permite la sustitución directa en la forma estándar para obtener la ecuación.Si la ecuación no está en forma estándar, agrupar términos y completar el cuadrado revela el centro y el radio del círculo.

círculos

Un c&#237;rculo en el plano de coordenadas se define como el conjunto de todos los puntos que est&#225;n a una distancia constante, conocida como radio, de un punto fijo llamado centro. Esta relaci&#243;n se representa mediante la f&#243;rmula de la distancia. Para un punto (x, y) en el c&#237;rculo y un centro (h, k), la distancia entre ellos es igual al radio r. Al elevar al cuadrado ambos lados de la f&#243;rmula de la distancia, la ecuaci&#243;n del c&#237;rculo se expresa en forma est&#225;ndar:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_1_1__editor_40041.svg' alt='40041_Equation_1_1_.svg' /></figure>Construcci&#243;n de la ecuaci&#243;n a partir de los datos geom&#233;tricosSi se conocen el centro y el radio de un c&#237;rculo, la ecuaci&#243;n se puede formular directamente mediante sustituci&#243;n. Sea un c&#237;rculo con centro en (1, 2) y radio 4. Al sustituir en la forma est&#225;ndar, se obtiene:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_2_1__editor_40041.svg' alt='40041_Equation_2_1_.svg' /></figure>Esta ecuaci&#243;n representa todos los puntos (x, y) que se encuentran a 4 unidades del punto (1, 2). Este enfoque sencillo suele ser suficiente para construir ecuaciones cuando se proporcionan expl&#237;citamente las caracter&#237;sticas geom&#233;tricas del c&#237;rculo.Identificaci&#243;n del c&#237;rculo a partir de una ecuaci&#243;n generalCuando la ecuaci&#243;n del c&#237;rculo no se presenta en forma est&#225;ndar, el procedimiento de completar el cuadrado permite identificar su centro y su radio. Considere la ecuaci&#243;n:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_3_1__editor_40041.svg' alt='40041_Equation_3_1_.svg' /></figure>Agrupe y complete el cuadrado:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_4_1__editor_40041.svg' alt='40041_Equation_4_1_.svg' /></figure>&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_5_1__editor_40041.svg' alt='40041_Equation_5_1_.svg' /></figure>Esto revela el centro en (3, 2) y un radio de 4. Expresar las ecuaciones en forma est&#225;ndar es esencial para representar y analizar c&#237;rculos eficazmente.

Círculos

Un c&#237;rculo en el plano de coordenadas se define como el conjunto de todos los puntos que est&#225;n a una distancia constante, conocida como radio, ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Coordinates and Graphs

Coordenadas y gráficas

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Circles

circle equations in standard form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius.

The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form.

If the center and radius are known, the equation can be built directly. For a circle with center 1, 2 and radius 4, substitute those values into the standard form. The result describes every point that is 4 units from 1, 2.

Sometimes a circle is given in a more general equation instead of standard form. In that case, completing the square helps reveal the center and radius. For example, grouping the terms and completing the square can show a center at 3, 2 and a radius of 4. Rewriting the equation in standard form makes circles easier to graph and analyze.

Circle Equations in Standard Form

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

Finding Distance on a Coordinate Plane

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings. The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them. <br...

the midpoint formula

Finding the Center of a Line Segment

Finding the center of a line segment is a key idea in coordinate geometry. The midpoint is the point that lies exactly halfway between two points on a coordinate plane. It is used in mathematics, physics, engineering, and planning work where a central location matters. To work with a midpoint, start with two points, A(x1, y1) and B(x2, y2). A straight line segment can be drawn between them on the coordinate plane. The midpoint, labeled M, divides that segment into two equal parts. The...

graphs of equations in two variables

Plotting Two-Variable Equations with Tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

An ellipse centered at the origin shows clear symmetry in its graph. Its equation describes a smooth, closed curve whose distance from the center keeps a constant ratio between the horizontal and vertical axes. Because the ellipse stretches farther along the x-axis than the y-axis, it has a horizontal shape. This ellipse has symmetry across the x-axis. That means the top half and bottom half are mirror images across the horizontal axis. Algebraically, the equation stays the same when y is...

solving equations graphically

Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

Graphing Inequality Solution Ranges

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Learn how to write a circle equation from a center and radius, and how completing the square reveals the center and radius from a general equation.

Circle Equations in Standard Form | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Circle Equations in Standard Form

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