Problem 64 - Thomas Calculus

Question

Find an equation for the set of all points equidistant from the point \((0,0,2)\) and the \(x y\) -plane.

Step-by-Step Solution

Answer

The equation is \(x^2 + y^2 - 4z + 4 = 0\).

Step 1: Understand the Problem

We need to find the equation for the set of all points that are equidistant from a given point, \((0,0,2)\), and the \(xy\)-plane. A point is represented by \((x, y, z)\) and we need to determine conditions under which the distance from \((x, y, z)\) to \((0, 0, 2)\) is the same as the distance to the \(xy\)-plane.

Step 2: Calculate the Distance to the Point

The distance between a general point \((x, y, z)\) and the point \((0, 0, 2)\) is given by the distance formula: \( \sqrt{(x-0)^2 + (y-0)^2 + (z-2)^2} = \sqrt{x^2 + y^2 + (z-2)^2} \).

Step 3: Calculate the Distance to the XY-Plane

The distance from the point \((x, y, z)\) to the \(xy\)-plane is simply \(|z|\), since only the \(z\)-coordinate determines distance to the \(xy\)-plane (\(x\) and \(y\) are unaffected as points on the plane).

Step 4: Equate the Distances

Set the two distances equal to each other:\[\sqrt{x^2 + y^2 + (z-2)^2} = |z|.\]

Step 5: Simplify the Equation

Square both sides to eliminate the square root and absolute value:\[x^2 + y^2 + (z-2)^2 = z^2.\]Expanding and simplifying gives:\[x^2 + y^2 + z^2 - 4z + 4 = z^2.\]Cancel \(z^2\) from both sides:\[x^2 + y^2 - 4z + 4 = 0.\]

Step 6: Adjust the Equation Form

The equation \(x^2 + y^2 - 4z + 4 = 0\) is the equation of the set of all points equidistant from the point \((0,0,2)\) and the \(xy\)-plane.

Key Concepts

Distance Formula

The distance formula is a fundamental concept in geometry that helps determine the distance between two points in space. It extends the Pythagorean theorem to calculate distances in more than one dimension. Here’s how it works: if you have two points, - Point A with coordinates \(x_1, y_1, z_1\) - Point B with coordinates \(x_2, y_2, z_2\) The distance between these points in 3D space is given by the formula: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]For our specific problem, we want the distance from \(x, y, z\) to \(0, 0, 2\). Substituting these values into the formula, we arrive at: \[ \sqrt{x^2 + y^2 + (z-2)^2} \]This formula helps us systematically find distances between any two points, whether they're on a single plane or spread out in three dimensions. The distance formula is crucial when you're dealing with problems that involve locating points in space or analyzing the relationship between various positions.

3D Geometry

3D geometry extends the concepts of 2D geometry into the third dimension, adding depth to the familiar length and width of a two-dimensional plane. This field of study is vital for understanding objects and spaces that, like our problem, exist in three dimensions. When dealing with 3D objects, students should consider "z", the vertical component, alongside the standard "x" and "y" horizontal components encountered in 2D geometry.

In our example, we consider the distance from a point in space to the \(xy\) plane, which is the horizontal plane where points have different "z" coordinates. This also involves computing tangible measurements over these 3 dimensions. - Point \(0, 0, 2\) exists at a height of "2" on the z-axis. - A general point \(x, y, z\) varies over all three axes.

The concept of equidistance in 3D occurs when a point maintains the same measurement from a fixed location to a virtual plane like the \(xy\) plane. Grasping the spatial relationship of points and planes helps decode problems related to depth, such as generating models and determining volumes.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, bridges algebra and geometry via coordinates to solve geometric problems. By using algebraic formulas, it expresses geometrical properties mathematically and thus makes calculations feasible for complex shapes and configurations.

In this setup, each geometric figure or line is identified and handled purely by its coordinates. For example: - The \(xy\) plane is defined where "z" is always zero.- A point \(0, 0, 2\) lies 2 units above or below the \(xy\) plane depending on perspective.- The expression \(\sqrt{x^2 + y^2 + (z-2)^2} = |z|\) emanates from applying analytical geometry techniques to equate distances.

In analytical geometry, the focus lies on using equations to derive the positions and conditions of existence for points and lines in space. The simplified equation \(x^2 + y^2 - 4z + 4 = 0\) in our problem is a perfect example of how combining algebraic manipulation with geometric insight can determine an equal-distance criteria for a set of points.