Question

In Exercises 45 and \(46,\) analyze the trace when the surface \(z=\frac{1}{2} x^{2}+\frac{1}{4} y^{2}\) is intersected by the indicated planes. Find the coordinates of the focus of the parabola formed when the surface is intersected by the planes given by (a) \(y=4 \quad\) and (b) \(x=2\).

Short answer

The focus of the parabola formed when the surface z = 1/2 * x^2 + 1/4 * y^2 is intersected by the plane y = 4 is at \((0, 4, 4 + \frac{1}{8})\) and when intersected by the plane x = 2 is at \((2, 0, 2 + \frac{1}{16})\).

Step-by-step solution

Intersecting Surface with plane y = 4

First, by substituting \(y = 4\) into the equation for the surface, we get \(z = \frac{1}{2} x^{2} + 4\). This is a parabola with its axis parallel to the z-axis and its focus is determined by the equation \(4p = \frac{1}{2}\) where \(p\) is the distance from the vertex to the focus.

Find the focus of the parabola z = 1/2 * x^2 + 4

Here, rearrange the equation: \(4p = \frac{1}{2}\). Solving for \(p\), we find \(p = \frac{1}{8}\). Hence, the coordinates of the focus in this case will be \((0, 4, 4 + \frac{1}{8})\).

Intersecting Surface with plane x = 2

Second, by substituting \(x = 2\) into the equation for the surface, we get \(z = 2 + \frac{1}{4} y^{2}\). This is also a parabola with its axis parallel to the z-axis.

Find the focus of the parabola z = 2 + 1/4 * y^2

From the equation for this parabola, we get \(4p = \frac{1}{4}\), so \(p = \frac{1}{16}\). Therefore, the coordinates of the focus will be \((2, 0, 2 + \frac{1}{16})\).

Key concepts

Parabola Focus Determination

Understanding the focus of a parabola is crucial in calculus, especially when dealing with quadratic surfaces. A parabola is the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, known as the directrix. To determine the focus of a parabola, we use the general equation of a parabola in the form of \( y = ax^{2} + bx + c \) for a vertical parabola or \( x = ay^{2} + by + c \) for a horizontal parabola.

The formula to find the focus from the vertex \( (h,k) \) is \( (h, k + \frac{1}{4a}) \) or \( (h + \frac{1}{4a}, k) \), depending on the orientation of the parabola. The value \( \frac{1}{4a} \) represents the distance \( p \) from the vertex to the focus, where \( a \) is the coefficient of the quadratic term. To find the focus from a standard equation, you first need to complete the square to bring it to the vertex form.

In the given problem, we're given the surface equation \( z = \frac{1}{2} x^{2} + \frac{1}{4} y^{2} \). When we intersect this surface with the plane by setting \( y = 4 \), we find that the equation simplifies to a parabola in \( x \) and \( z \). By comparing this equation to the standard form, we can use the relation \( 4p = \frac{1}{2} \) to find the focus.

Intersecting Planes with Surfaces

The idea of intersecting a plane with a surface in 3D space is a fundamental concept in calculus and is used to comprehend how two-dimensional figures can be sliced from three-dimensional objects. Intersections usually create a locus of points forming a curve or a shape, such as lines, circles, ellipses, or parabolas, depending on the original surface and the intersecting plane.

Given a quadratic surface, such as \( z = \frac{1}{2} x^{2} + \frac{1}{4} y^{2} \), and a plane (e.g., \( y = 4 \) or \( x = 2 \)), you substitute the equation of the plane into the surface to find the intersection curve. This operation effectively reduces one variable, leading to a 2D equation that is easier to study.

The steps typically involve:

• Substituting the value from the plane's equation into the surface's equation.
• Reducing the resulting equation to represent a curve in two dimensions.
• Analyzing the reduced equation to identify the type of curve and its properties.

Quadratic Surfaces

Quadratic surfaces are 3D analogs of conic sections and appear frequently in mathematics, physics, and engineering. They include paraboloids, ellipsoids, hyperboloids, and cylinders. These surfaces are defined by second-degree equations in three variables: \( x \), \( y \), and \( z \).

The equation \( z = \frac{1}{2} x^{2} + \frac{1}{4} y^{2} \) represents an elliptic paraboloid, a type of quadratic surface. It's formed by shifting a parabola along an elliptical path. This specific shape is convex and opens upward because the coefficients of \( x^{2} \) and \( y^{2} \) are both positive.

When we intersect a quadratic surface with a plane, we can obtain various sections, most of which are conic sections themselves. In the case of an elliptic paraboloid, intersecting it with a horizontal plane will result in an ellipse, while a vertical plane may result in a parabola or a hyperbola, depending on the angle and location of the intersection.