Problem 43 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) ASSORTED $$4 y^{2}+z^{2}-4 x^{2}=4$$

Step-by-Step Solution

Answer

The surface is a hyperboloid of one sheet centered at the origin.

Step 1: Identify the Type of Surface

The given equation is: \[ 4y^2 + z^2 - 4x^2 = 4 \]. This equation resembles the standard form of a hyperboloid. Notice that it has both positive and negative signs in the quadratic terms, indicating it is a hyperboloid.

Step 2: Simplify the Equation

Divide the entire equation by 4 to simplify it and compare it to the standard form of a hyperboloid:\[ \frac{y^2}{1} + \frac{z^2}{4} - \frac{x^2}{1} = 1 \]. This helps to identify the surface type clearly, indicating a hyperboloid of one sheet.

Step 3: Analyze the Surface's Axes

The equation \( \frac{y^2}{1} + \frac{z^2}{4} - \frac{x^2}{1} = 1 \) indicates the hyperboloid is centered at the origin \((0,0,0)\). The \(x^2\) term is negative, meaning the surface opens along the y-axis and z-axis.

Step 4: Consider the Cross Sections

If we set \(x = c\), where \(c\) is a constant, we get ellipses in the yz-plane. For \(y = k\) or \(z = m\), if constant planes contain segments not forming closed loops (due to the minus term), mapping hyperbolic-like curves can be expected.

Step 5: Visualize the Surface

Imagine the surface extending along the positive and negative y, and z directions with an ellipse cross-section parallel to the yz-plane. This visual helps in sketching the general shape of the hyperboloid.

Key Concepts

Quadratic Surfaces

Quadratic surfaces are three-dimensional surfaces represented by quadratic equations. These mathematical surfaces include a variety of well-known shapes like ellipsoids, paraboloids, and hyperboloids. Quadratic surfaces take the general form:
\[ Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fzx + Gx + Hy + Iz + J = 0 \]
where each letter represents constants that shape the surface.

• Ellipsoids have all positive quadratic terms and form closed shapes like spheroids.
• Paraboloids have mixed signs and form open surfaces similar to parabolas revolved in space.
• Hyperboloids can have one or two sheets, depending on the signs and coefficients of the terms, leading to open surfaces that might extend indefinitely.

For a hyperboloid of one sheet, seen from the equation
\[ 4y^2 + z^2 - 4x^2 = 4 \],
you notice the quadratic terms have positive and negative signs. This characteristic sets hyperboloids apart in quadratic surfaces, distinguishing them by their unique asymptotic behaviors.

Coordinate Geometry

Coordinate geometry, or analytic geometry, bridges algebraic equations with geometric figures. By using coordinates, you can depict complex surfaces like the hyperboloid via simpler mathematical forms. The transformation of the quadratic equation
\[ 4y^2 + z^2 - 4x^2 = 4 \]
into its simplified form
\[ \frac{y^2}{1} + \frac{z^2}{4} - \frac{x^2}{1} = 1 \]
is a perfect example of how coordinate geometry assists in visualizing and understanding geometrical shapes.

• The surface is centered at the origin \((0,0,0)\), highlighting the symmetry in all directions.
• The equation reveals the directions (axes) along which the surface extends or opens. For hyperboloids, positive terms show expanding directions, while the negative term's direction often acts as an imaginary axis.

Coordinate geometry helps explore how shapes stretch, transform, and relate to each other in three-dimensional space.

Cross Sections

Cross sections involve slicing a three-dimensional object with a plane, revealing shapes that lie within. For a hyperboloid of one sheet like the one described by \[ \frac{y^2}{1} + \frac{z^2}{4} - \frac{x^2}{1} = 1 \],
different cross sections can reveal ellipses or hyperbolas, depending on the orientation of the cut.

• Setting \(x = c\), a constant, gives elliptical cross-sections in the yz-plane.
• For cuts where \(y = k\) or \(z = m\), the sections may form hyperbolic curves or segments, reflecting the asymptotic nature of hyperboloids.

These cross sections highlight the underlying structure and can clarify how the equations relate to visual geometry. Cross sections are useful as they provide insight into the interior form without needing the complete volume view, often utilized in engineering and design fields.