From the focus/directrix definition of a hyperbola: If the...

From the focus/directrix definition of a hyperbola: If the distance from the focus to a point $(x, y)$ is greater than the distance from the directrix to $(x, y)$ one branch of a hyperbola is formed. Using (2,0) as the focus and the vertical line $x=\frac{1}{2}$ as the directrix, find an equation for the set of all points $(x, y)$ where the distance from the focus to $(x, y),$ is twice the distance from the directrix to $(x, y)$.

From the focus/directrix definition of a hyperbola: If the distance from the focus to a point $(x, y)$ is greater than the distance from the directrix to $(x, y)$
one branch of a hyperbola is formed. Using (2,0) as the focus and the vertical line $x=\frac{1}{2}$ as the directrix, find an equation for the set of all points $(x, y)$ where the distance from the focus to $(x, y),$ is twice the distance from the directrix to $(x, y)$.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone, characterized by specific relationships between distances from any point on the curve to a fixed point (the focus) and to a fixed line (the directrix). These curves include ellipses, parabolas, and hyperbolas, which differ based on their eccentricity values.

Focus and Directrix

The focus is a fixed point, and the directrix is a fixed line used in the geometric definition of conic sections. A conic is defined as the set of points for which the distance to the focus is a constant multiple (the eccentricity) of the distance to the directrix.

Eccentricity

Eccentricity is a constant that describes the shape of a conic section. For hyperbolas, the eccentricity is greater than 1, meaning that the distance from any point on the hyperbola to the focus is a specified multiple (greater than 1) of its distance to the directrix.

Distance Formulas

Distance formulas are essential tools in coordinate geometry. The Euclidean distance formula is used to compute the distance between two points, while the formula for the distance from a point to a line calculates the shortest distance from a given point to a specified line. These formulas are crucial when deriving the equations of conic sections from their geometric definitions.

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