From the two foci definition of a hyperbola: For foci f1 and...

From the two foci definition of a hyperbola: For foci $f_{1}$ and $f_{2},$ a hyperbola is the set of all points $(x, y)$ where the difference of the distances from $f_{1}$ to $(x, y)$ and $f_{2}$ to $(x, y)$ is constant. Verify the points (2,3) and $(-3,-2 \sqrt{6})$ are on the graph of the hyperbola from Exercise $33 .$ Then verify $d_{1}-d_{2}=d_{3}-d_{4}$ (GRAPH CAN'T COPY)

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Key Concepts

Hyperbola Definition

A hyperbola is defined as the set of all points in a plane such that the absolute difference of the distances from any point to two fixed points (called the foci) is constant. This fundamental definition distinguishes hyperbolas from other conic sections and is key to understanding their geometric properties.

Foci

The foci are two specific fixed points that, along with the constant distance difference condition, determine the shape and position of a hyperbola. Their locations relative to each other affect the hyperbola's eccentricity and overall orientation in the plane.

Distance Formula

The distance formula, based on the Pythagorean theorem, is used to calculate the distance between two points in a coordinate plane. This tool is critical for verifying the hyperbola's defining property by measuring distances from various points to the foci.

Verification of Points on a Hyperbola

Verifying whether a given point lies on a hyperbola involves calculating its distances to the foci using the distance formula and checking that the absolute difference of these distances equals the prescribed constant. This process is a direct application of the hyperbola's definition.

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