Question

Find the equation of the surface that results when the curve \(4 x^{2}-3 y^{2}=12\) in the \(x y\) -plane is revolved about the \(x\) -axis.

Short answer

The equation of the surface is \(z^{2} + y^{2} = \frac{4x^{2} - 12}{3}\).

Step-by-step solution

Identify the Curve

The given curve is a hyperbola, represented by the equation \(4x^{2} - 3y^{2} = 12\). This is a horizontal hyperbola since the \(x^{2}\) term is positive.

Rearrange and Standardize

First, rearrange the given hyperbola equation into its standard form by dividing all terms by 12:\[ \frac{x^{2}}{3} - \frac{y^{2}}{4} = 1 \]This is the standard form of the hyperbola centered at the origin, opening sideways.

Set Up the Revolution About the x-axis

When the curve is revolved around the \(x\)-axis, each point on the hyperbola forms a circle in the \(yz\)-plane. Thus, for every \((x, y)\) point on the curve, this gives rise to a circle equation in the \(yz\)-plane:\[ z^{2} + y^{2} = r^{2} \]Here, \(r\) is the distance from the curve to the \(x\)-axis, which is simply \(|y|\). Thus, \(z^{2} + y^{2} = y^{2}\).

Formulate the Surface Equation

To find the equation of the surface, substitute \(y^{2}\) from the hyperbola's equation into the circular cross-section equation:1. From the rearranged hyperbola equation: \(y^{2} = \frac{4x^{2} - 12}{3}\).2. Substitute \(y^{2}\) in the circle equation:\[ z^{2} + \frac{4x^{2} - 12}{3} = \frac{4x^{2} - 12}{3} \]This simplifies to \(z^{2} + y^{2} = \frac{4x^{2} - 12}{3}\), and thus provides the full surface equation as it's not dependent on \(z\) any more.

Key concepts

Hyperbola

Understanding the concept of a hyperbola is crucial for approaching problems like our original exercise. A hyperbola is a type of conic section that appears as two separate curves opening in opposite directions. It arises when a plane cuts through both nappes of a double cone but not through its base. The defining feature of a hyperbola is its unique equation, generally expressed in the form:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

This form allows us to tell that the hyperbola in question is horizontally oriented because the \(x^2\) term comes first. The standard form gives insight into its geometry, such as the distances to vertices and asymptotes. For the problem at hand, recognizing that our given curve \(4x^2 - 3y^2 = 12\) is a hyperbola is essential to finding the surface equation upon revolution.

Revolution about the axis

When we talk about revolution in geometry, we're referring to the concept of taking a 2D shape and rotating it around an axis to create a 3D object. In our exercise, we revolve the hyperbola around the x-axis. This forms a three-dimensional shape because every point on the hyperbola traces out a circle as it rotates around the axis. Imagine a spinning wheel turned sideways; every spoke traces a circle perpendicular to the spin axis. The mathematical result is a surface of revolution. This process has applications in engineering and physics, where objects of symmetrical shape are studied for various properties.

Surface Equation

To describe the new three-dimensional shape formed by revolving a curve, we need to formulate the surface equation. Here’s how we achieved that in the original problem:

• We first rearranged the given hyperbola equation into standard form: \( \frac{x^2}{3} - \frac{y^2}{4} = 1 \).
• When revolving about the x-axis, any point \( (x, y) \) yields a circle in the yz-plane with the equation \( z^2 + y^2 = r^2 \), where \( r = |y| \).

To form the composite structure, we substitute \( y^2 \) obtained from the hyperbola into this circular equation. Thus, the surface equation derived was \( z^2 = \frac{4x^2 - 12}{3} \), completing the surface's description as a function of \( x \) and \( z \). This formulation method offers a systematic pathway from simple curves to complex surfaces.

Coordinate Geometry

Coordinate geometry is a crucial mathematical area that combines algebra and geometry, allowing us to solve problems about lines, curves, and surfaces efficiently. In the exercise, we used coordinate geometry to seamlessly transition from a given equation of a hyperbola to its standard form, enabling straightforward calculations for the resulting solid of revolution. This discipline often deals with different coordinate systems such as Cartesian and polar, but here, we operate in Cartesian coordinates, leveraging the coordinate plane to interpret shapes and their equations effectively. This method is the bridge that translates geometric intuition into precise equations that describe our world mathematically. Understanding these fundamental concepts is key to tackling a variety of mathematical and real-world applications.