Problem 45 - Calculus with Concepts in Calculus

Question

Sketch the quadric surface. \(x^{2}=9 y^{2}+4 z^{2}\)

Step-by-Step Solution

Answer

The equation represents an elliptic cone centered at the origin and opening along the x-axis.

Step 1: Rewrite the Equation

The given equation is \(x^2 = 9y^2 + 4z^2\). This can be rewritten in a more recognizable form for quadric surfaces by dividing every term by 36 (which is the least common multiple of 9 and 4): \(\frac{x^2}{36} = \frac{y^2}{4} + \frac{z^2}{9}\).

Step 2: Identify the Surface

The rewritten equation \(\frac{x^2}{36} = \frac{y^2}{4} + \frac{z^2}{9}\) is in the standard form of an elliptic cone. In general, an equation of the form \(\frac{x^2}{a^2} = \frac{y^2}{b^2} + \frac{z^2}{c^2}\) represents a cone with its axis parallel to the x-axis.

Step 3: Determine Cross Sections

To sketch the surface, start by determining its cross-sections. If you set \(x = k\), where \(k\) is a constant, the equation becomes \(k^2 = 9y^2 + 4z^2\). This represents an ellipse in the yz-plane with semi-major axis \(\frac{k}{3}\) and semi-minor axis \(\frac{k}{2}\).

Step 4: Sketch the Surface

Draw the y-axis and z-axis on standard graph paper. For different values of \(x\) (e.g., \(x = 0, \pm 1, \pm 2, \ldots\)), draw the corresponding ellipses determined in Step 3. Ensure the ellipses increase in size as \(|x|\) increases. This forms a cone opening along the x-axis, with its vertex at the origin (0,0,0).

Step 5: Verify Symmetry

Note the symmetry of the surface: the equation is symmetric concerning the yz-plane (if x is replaced with -x, the equation remains unchanged), confirming it is symmetric about the yz-plane along the x-axis.

Key Concepts

Elliptic Cone

An elliptic cone is a type of quadric surface that is characterized by its conical shape with an elliptical cross-section. In mathematical terms, an \[ \frac{x^2}{a^2} = \frac{y^2}{b^2} + \frac{z^2}{c^2} \] represents an elliptic cone, with the axis of the cone parallel to the x-axis, which is the direction along which the cone opens. This particular equation emphasizes how the elliptical nature comes into play due to the terms on the right-hand side, which define an ellipse in the y-z plane. Since the entire surface extends infinitely along the x-axis, it forms a double-cone structure, with one part going in the positive and the other in the negative x-direction. It's this characteristic that distinguishes it as an elliptic cone, as opposed to other types like circular cones.

Cross-sections

To understand the shape of an elliptic cone, we need to consider its cross-sections, which are obtained by slicing the cone perpendicular to its axis. For an elliptic cone given by the equation \[ x^2 = 9y^2 + 4z^2,\] we can analyse the cross-sections by setting the x-value to a constant. When we set \( x = k \), the resulting equation becomes \[ k^2 = 9y^2 + 4z^2.\] This describes an ellipse in the yz-plane, where the lengths of the semi-axes are related to the constant k. Specifically, the semi-major axis length will be \( \frac{k}{3} \) and the semi-minor axis length will be \( \frac{k}{2} \). Thus, as \ x gets larger, the ellipses get larger as well; this collection of growing ellipses along the x-axis forms the cone. Observing these cross-sections provides a clear visualization of the cone’s changing shape at various points along its axis.

Symmetry

Symmetry is a key property of many mathematical surfaces, including elliptic cones. For the equation \[ x^2 = 9y^2 + 4z^2, \] symmetry can be verified by examining how the equation behaves when we swap or negate its variables. Notice that if we replace \( x \) with \( -x \), the equation remains unchanged: \[ (-x)^2 = 9y^2 + 4z^2, \] which confirms that the surface is symmetric with respect to the yz-plane. This means that the cone is mirrored on either side of the yz-plane, reflecting the geometric property of symmetry about its axis of rotation, which extends horizontally along the x-axis. Such symmetry helps simplify the process of sketching and understanding the geometry of the surface, as it reduces the complexity of the surface without losing information about its shape.

Standard Form

An equation is considered to be in standard form when it is arranged in a manner that highlights certain features of the geometric figure it represents. For quadric surfaces like an elliptic cone, the standard form is crucial as it provides insights into the orientation and dimensions of the surface. The equation \[ x^2 = 9y^2 + 4z^2 \] is not in standard form initially. To convert it, we divide each term by 36 (the least common multiple of 9 and 4) to get:\[ \frac{x^2}{36} = \frac{y^2}{4} + \frac{z^2}{9}. \]This step is essential because it aligns the equation with the conventional representation of an elliptic cone, clearly indicating that the cone extends along the x-axis, with its vertex at the origin. Having the equation in standard form simplifies the process of identifying and analyzing the geometric properties of the surface, such as locating symmetry, axes, and cross-sectional shapes.