Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the coordinates of the focus of the parabola that is the intersection of \(z=x^{2} / 4+y^{2} / 9\) with \(x=4\).

Step-by-Step Solution

Answer

The focus is at \(\left(4, 0, \frac{25}{4}\right)\).

Step 1: Understanding the Problem

We need to find the coordinates of the focus of the parabola formed by the intersection of the surface \(z = \frac{x^2}{4} + \frac{y^2}{9}\) with the plane \(x = 4\). This means substituting \(x = 4\) into the equation and analyzing the result as a parabola.

Step 2: Substituting x = 4

Substitute \(x = 4\) into the equation \(z = \frac{x^2}{4} + \frac{y^2}{9}\). This gives us \(z = \frac{4^2}{4} + \frac{y^2}{9} = 4 + \frac{y^2}{9}\). So, the equation of the parabola is \(z = 4 + \frac{y^2}{9}\).

Step 3: Reformulating the Parabola Equation

Since we need the parabola in standard form, rearrange the equation into \(z - 4 = \frac{y^2}{9}\). This can be written as \((z - 4) = \frac{1}{9}y^2\), which matches the standard parabola form \(z - c = \frac{1}{4p}(y-k)^2\). Here, \(c = 4\) and \(k = 0\).

Step 4: Finding the Focus

The standard form \((z - c) = \frac{1}{4p}(y-k)^2\) implies that \(4p = 9\), so \(p = \frac{9}{4}\). The focus of a parabola is at \((c + p, k)\), therefore the focus is at \((4 + \frac{9}{4}, 0)\).

Step 5: Calculating the Coordinates

Calculate the focus's coordinates. Since \(c = 4\), and \(p = \frac{9}{4}\), the focus \((c + p, k)\) becomes \((4 + \frac{9}{4}, 0) = (\frac{16}{4} + \frac{9}{4}, 0) = (\frac{25}{4}, 0)\). Thus, in three dimensions the focus is at \((\frac{25}{4}, 0, 4)\) on the original coordinate plane with \(x = 4\).

Key Concepts

Coordinate geometry

Coordinate geometry is the study of geometric figures through algebra using coordinate points. It involves understanding shapes and their properties within a coordinate system, like the x, y, and z axes in 3D space.
Coordinate geometry helps in

• Analyzing geometric shapes
• Finding distances between points
• Slope calculations for lines
• Intersections of different geometric figures

By applying formulas like distance or midpoint, it becomes easier to solve geometry problems in a more algebraic way.

3D intersection

In problems involving spatial figures, a 3D intersection refers to the overlapping or meeting of two or more objects in three-dimensional space.
To find intersections:

• Identify the equations representing the surfaces or objects involved
• Substitute variables to reduce dimensions, if needed
• Solve resulting equations to find intersection points

Here, the parabola is discovered by evaluating the intersection of the surface described by the equation \(z = \frac{x^2}{4} + \frac{y^2}{9}\) with the plane \(x = 4\). This substitution reduces the problem to two dimensions, where the intersection forms a parabola.

Parabola standard form

The standard form of a parabola's equation is crucial when identifying its axis of symmetry, vertex, and focus.
A common form for parabolas in 2D is

• \((z - c) = \frac{1}{4p}(y-k)^2\),

where

• \(c\) and \(k\) are constants representing the coordinates of the vertex,
• \(p\) is the distance from the vertex to the focus.

By rearranging your original equation \((z - 4) = \frac{y^2}{9}\), we determine that the equation fits the pattern of a parabola opening along the z-axis, giving direct insight into its focus and vertex.

Analytic geometry

Analytic geometry involves using algebra to solve geometric problems by representing figures using equations. It combines elements of algebra and geometry, allowing us to describe shapes in coordinate systems.
Key aspects include:

• Using coordinates to represent geometric figures
• Solving equations to determine attributes like intercepts, slopes, and intersections
• Graphing equations to visualize and interpret solutions

By formulating the equation of the surface and intersections as mathematical expressions, analytic geometry helps translate spatial relationships into algebraic equations that are more straightforward to manipulate.