Problem 31 - Essential Calculus Early Transcendentals

Question

Find an equation for the surface consisting of all points that are equidistant from the point \((-1,0,0)\) and the plane \(x=1 .\) Identify the surface.

Step-by-Step Solution

Answer

The surface is a paraboloid with the equation \(y^2 + z^2 = -4x\), opening along the negative x-axis.

Step 1: Define the Distance to a Point

The distance from a point \((x, y, z)\) to the point \((-1, 0, 0)\) can be calculated using the distance formula. It is given by:\[\sqrt{(x + 1)^2 + y^2 + z^2}\]

Step 2: Define the Distance to a Plane

The distance from a point \((x, y, z)\) to the plane defined by \(x = 1\) is the absolute value of the difference between the x-coordinate, \(x\), and 1. Thus, the distance is given by:\[|x - 1|\]

Step 3: Set Distances Equal

A point is equidistant from \((-1, 0, 0)\) and the plane \(x=1\) if its distance to each is equal. Therefore, we set the distances calculated in Steps 1 and 2 equal to each other:\[\sqrt{(x + 1)^2 + y^2 + z^2} = |x - 1|\]

Step 4: Square Both Sides

To eliminate the square root, square both sides of the equation from Step 3:\[(x + 1)^2 + y^2 + z^2 = (x - 1)^2\]

Step 5: Expand and Simplify

Expand the squared terms:\[(x^2 + 2x + 1) + y^2 + z^2 = x^2 - 2x + 1\]Simplify by subtracting \(x^2 + 1\) from both sides:\[2x + y^2 + z^2 = -2x\]

Step 6: Solve for y and z Terms

Move the \(2x\) term to the other side:\[y^2 + z^2 = -4x\]Divide by \(-1\) to rearrange the equation in a standard form:\[y^2 + z^2 = -4x\]

Step 7: Identify the Surface

The equation obtained resembles that of a paraboloid, which is defined as being symmetric around an axis (here the x-axis) and extends infinitely in one direction. The form \(y^2 + z^2 = -4x\) shows it to be a paraboloid opening along the negative x-axis.

Key Concepts

Distance to a Point

In mathematics, the distance from one point to another is a measure of how far apart the two points are. The distance from a point \((x, y, z)\) to a fixed point, such as \((-1,0,0)\), can be calculated using the three-dimensional distance formula. This formula is given as:\[ \sqrt{(x + 1)^2 + y^2 + z^2} \]This formula comes from the Pythagorean theorem, extended into three dimensions. It calculates the straight-line (or "Euclidean") distance between two points in space. Here, you add 1 to the x-coordinate because the fixed point has an x-value of -1. Every part of this formula represents component-wise differences raised to power of two. This ensures that the distance measure remains positive, regardless of the position of the points in the three-dimensional space.

Distance to a Plane

The concept of distance to a plane involves finding how far a point is from a given plane. Here, our plane is described simply by the equation \(x = 1\). Planes in three-dimensional space often involve a more complex relationship between x, y, and z coordinates, but this one is straightforward.For a point \((x, y, z)\) in space, the distance to the plane \(x = 1\) is the absolute value of the x-coordinate minus 1, which is represented by:\[ |x - 1| \]This equation reflects the direct manner in which vertical (or horizontal) distances between a point and the plane are calculated. Here, since \(x = 1\) represents a vertical plane perpendicular to the x-axis, the distance is purely determined by differences in the x-coordinate. No y or z components are required because the plane is independent of these dimensions.

Paraboloid

A paraboloid is a three-dimensional surface that looks somewhat like an infinite bowl shape. Its cross-section is a parabola, which gives it its name. In this exercise, we obtained the equation \(y^2 + z^2 = -4x\) after manipulating the distances.The equation here shows a paraboloid symmetric about the x-axis. This is because both \(y^2\) and \(z^2\) are involved, showing no preference in their dimension relative to one another.This specific equation represents a paraboloid opening along the negative x-axis, because the x-term is linear and has negative multiplication with constant.Key characteristics of paraboloids include:

• They have a vertex where they are most "narrow" or "closed." For our paraboloid, it is at the origin (0,0,0).
• When elongated infinitely in the positive or negative direction along its axis (in this case, the x-axis), it forms an open-ended surface.
• Each cross-section parallel to its base (in this case, perpendicular to the y-z plane) forms a circle.

Understanding paraboloids is essential for visualizing how geometric shapes can describe physical phenomena, like satellite dishes or cooling towers, both of which are practical applications of paraboloid shapes.