Problem 32 - Essential Calculus Early Transcendentals

Question

Find an equation for the surface consisting of all points \(P\) for which the distance from \(P\) to the \(x\) -axis is twice the distance from \(P\) to the \(y z\) -plane. Identify the surface.

Step-by-Step Solution

Answer

The surface is a double-napped cone with equation \(y^2 + z^2 = 4x^2\).

Step 1: Understand Point Definitions

A point \(P\) in 3D-space is typically defined as \((x, y, z)\). The distance from a point to the \(x\)-axis is the perpendicular distance to this line, which is equivalent to the distance within the \(y z\)-plane.

Step 2: Determine Distance Formulas

For a point \(P(x, y, z)\), the distance to the \(x\)-axis is \(\sqrt{y^2 + z^2}\). The distance to the \(y z\)-plane is \(|x|\).

Step 3: Set Up the Distance Equation

According to the problem statement, the distance to the \(x\)-axis is twice the distance to the \(y z\)-plane. Thus, we have:\[\sqrt{y^2 + z^2} = 2|x|\]

Step 4: Simplify the Equation

Square both sides of the equation to eliminate the square root:\[y^2 + z^2 = 4x^2\]

Step 5: Identify the Surface

The equation \(y^2 + z^2 = 4x^2\) represents a double-napped cone with its axis along the \(x\)-direction. This is because the typical form of a cone is \(y^2 + z^2 = c x^2\), and in this case, \(c = 4\).

Key Concepts

Distance in 3D Space

In three-dimensional space, understanding distance is crucial for analyzing spatial relationships. A point, represented as \(P(x, y, z)\), can have distances measured relative to various lines, planes, or points within the 3D coordinate system.

The distance from a point \(P\) to the \(x\)-axis is determined by projecting \(P\) onto the \(yz\)-plane. The formula is \(\sqrt{y^2 + z^2}\), derived from the Pythagorean theorem, as the distance is the hypotenuse of a right triangle formed by the \(y\) and \(z\) coordinates.

Similarly, the distance to the \(yz\)-plane, which is a vertical plane through the \(y\) and \(z\)-axis, is simply the absolute value of the \(x\)-coordinate, \(|x|\). This encapsulates the concept of projection along a perpendicular line, thus simplifying the spatial understanding in 3D space.

Equation of a Cone

A cone in 3D coordinate geometry is typically described by an equation similar to \(y^2 + z^2 = c x^2\), where \(c\) is a constant. This equation represents a double-napped cone. Each side of the cone opens away symmetrically along the central axis, which can be in any direction depending on the equation.

In the exercise, the derived equation \(y^2 + z^2 = 4x^2\) specifies a cone that is aligned with the \(x\)-axis. The factor 4 indicates how wide the cone opens; the larger the factor, the wider the cone. Understanding this structure is fundamental in visualizing how changes to parameters affect the resulting geometric figures.

With conic sections, transformations and rotations can further diversify the scope of shapes, making conic sections versatile and prevalent in various applications, from physics to engineering.

3D Coordinate Geometry

3D Coordinate Geometry extends the concepts of plane geometry into three-dimensional space, offering a richer framework for solving geometric problems. In 3D, points are defined by three coordinates \((x, y, z)\), providing a comprehensive positional system.

In solving problems, 3D coordinate geometry combines mathematical and spatial reasoning. By applying formulas like distances to lines and planes, and equations such as those representing cones and spheres, we can tackle a broad range of geometric queries.

Exploring 3D shapes through equations serves to not only understand their spatial properties but also to apply these properties in real-world problem-solving scenarios, such as visualizing objects, calculating volumes, or modeling physical phenomena. This makes 3D coordinate geometry an essential part of high school and higher education mathematics.