Find an equation of the surface consisting of all points...

Key Concepts

Deriving Equations through Algebraic Manipulation

Once the distance conditions are established, algebraic manipulation is used to transform and simplify the equations. This includes squaring both sides to eliminate square roots, expanding algebraic expressions, and simplifying by canceling common terms. Such techniques are fundamental in translating geometric definitions into their algebraic representations, which can be further analyzed or classified.

Focus-Directrix Definition and Parabolic Surface

A key concept in understanding the surface described is the focus-directrix property of a paraboloid, which is an extension of the parabola's definition from two dimensions to three dimensions. For a paraboloid, every point on the surface is equidistant from a fixed point known as the focus and a fixed plane known as the directrix. Recognizing this property allows one to derive the equation of the paraboloid by equating the distance from any point on the surface to the focus with the distance to the directrix plane.

Distance Formula in Three Dimensions

The distance formula in three-dimensional space is used to compute the distance between any two points (x1, y1, z1) and (x2, y2, z2). It is given by the square root of the sum of the squares of the differences in each coordinate, specifically ?[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]. This formula is fundamental for establishing relationships between spatial points and is essential when setting up equations based on distance conditions.

Distance from a Point to a Plane

The distance from a point to a plane in three-dimensional space is calculated using a formula that involves the coordinates of the point and the coefficients of the plane's equation. It is given by the absolute value of the linear expression obtained by substituting the point's coordinates into the plane's equation, divided by the square root of the sum of the squares of the plane's normal vector components. This concept is crucial when determining how closely a point lies relative to a given plane.

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