Using (0,-2) as the focus and the horizontal line y=-8 as...

Using (0,-2) as the focus and the horizontal line $y=-8$ as the directrix, find an equation for the set of all points $(x, y)$ where the distance from the focus to $(x, y)$ is one-half the distance from the directrix to $(x, y)$.

Key Concepts

Eccentricity

Eccentricity is a constant that defines the shape of a conic section by representing the ratio between the distance from any point on the conic to the focus and its distance to the directrix. Different values of eccentricity correspond to different conic sections: less than one for ellipses, exactly one for parabolas, and greater than one for hyperbolas.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry that calculates the distance between two points in the plane using their coordinates. It is crucial for deriving the equation of a conic section from the focus-directrix definition, as it allows the distance from any point on the curve to the focus to be equated with a constant multiple of the distance to the directrix.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They are defined by a fixed point called the focus and a fixed line called the directrix, along with a constant ratio known as the eccentricity. The type of conic—ellipse, parabola, or hyperbola—is determined by the value of the eccentricity.

Focus and Directrix

The focus is a specific point used in the geometric definition of a conic section, while the directrix is a fixed line. The conic is the set of all points for which the distance to the focus is a constant multiple (the eccentricity) of the distance to the directrix. This relationship lays the foundation for deriving the equation of the conic.

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