Analytic geometry

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Branch of mathematics

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Analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is widely used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight.

History

The field of analytic geometry was first developed by René Descartes and Pierre de Fermat in the 17th century. Descartes' work, "La Géométrie," laid the foundation for the use of algebra to describe geometric principles. Fermat's work on the same subject was published posthumously and contributed significantly to the development of the field.

Basic Concepts

Analytic geometry involves the use of the Cartesian coordinate system to describe geometric shapes. The Cartesian coordinate system is defined by a pair of perpendicular axes: the x-axis and the y-axis. Points in this system are described by ordered pairs \((x, y)\).

Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the plane can be found using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Midpoint Formula

The midpoint of the line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Slope of a Line

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Equations of Lines and Curves

In analytic geometry, lines and curves can be represented by equations.

Equation of a Line

The general form of the equation of a line is: \[ Ax + By + C = 0 \] where \(A\), \(B\), and \(C\) are constants.

The slope-intercept form of the equation of a line is: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.

Conic Sections

Conic sections, such as ellipses, parabolas, and hyperbolas, can also be described using equations in analytic geometry.

• The equation of a circle with center \((h, k)\) and radius \(r\) is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

• The equation of a parabola with vertex at the origin and axis of symmetry along the y-axis is:

\[ y = ax^2 \]

• The equation of an ellipse with center at the origin is:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

• The equation of a hyperbola with center at the origin is:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Applications

Analytic geometry is fundamental in various fields such as physics, engineering, and computer graphics. It is used to model and solve problems involving distances, midpoints, slopes, and intersections of geometric shapes.

Related Pages

• Geometry
• Cartesian coordinate system
• René Descartes
• Pierre de Fermat
• Conic section
• Distance formula
• Slope

See Also

• Synthetic geometry
• Euclidean geometry
• Non-Euclidean geometry
• Vector space