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How does an ordered pair identify a point in Cartesian coordinates?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, an ordered pair of numbers, denoted as $(x, y)$ , identifies the precise location of a point within a two-dimensional Cartesian coordinate system.1

This system relies on two perpendicular axes that intersect at a reference point called the origin, which is designated as $(0, 0)$ . The ordered pair functions as follows:21

The first number ( $x$ ): Represents the point's directed distance from the origin along the horizontal $x$ -axis.13
The second number ( $y$ ): Represents the point's directed distance from the origin along the vertical $y$ -axis.13

By measuring these distances from the origin, the coordinates provide a unique address for any point on the plane. This method allows for the algebraic representation of geometric shapes and positions, forming the foundation for solving complex problems in geometry through algebra.21