Question

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

Short answer

The foci of the ellipse are at \((0, \pm\sqrt{5})\) in the xy-plane.

Step-by-step solution

Understand the Problem

We are given an elliptical cylinder described by the equation \( z = \frac{x^2}{4} + \frac{y^2}{9} \), and we need to find the intersection of this cylinder with the plane \( z = 4 \). This intersection will form an ellipse.

Set Up the Equation for the Intersection

To find the intersection, set \( z = 4 \) in the given equation: \[ 4 = \frac{x^2}{4} + \frac{y^2}{9} \]. Multiply through by 36 to clear the fractions, resulting in the new equation: \[ 36 = 9x^2 + 4y^2 \]. This equation represents an ellipse in the \(xy\)-plane.

Write Equation in Standard Ellipse Form

The equation \( 36 = 9x^2 + 4y^2 \) can be rewritten in its standard form by dividing all terms by 36: \[ 1 = \frac{9x^2}{36} + \frac{4y^2}{36} \] which simplifies to \[ 1 = \frac{x^2}{4} + \frac{y^2}{9} \]. This indicates an ellipse centered at the origin with semi-major axis along the \( y\)-axis.

Determine Semi-Axes

From the equation \( 1 = \frac{x^2}{4} + \frac{y^2}{9} \), the semi-major axis is along the \( y\)-direction with length 3 (since \( \sqrt{9} = 3 \)), and the semi-minor axis is along the \( x\)-direction with length 2 (since \( \sqrt{4} = 2 \)).

Calculate the Foci

For an ellipse \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), the distance \( c \) from the center to each focus is given by \( c = \sqrt{a^2 - b^2} \). Substitute \( a = 3 \) and \( b = 2 \) into the formula: \[ c = \sqrt{3^2 - 2^2} = \sqrt{9 - 4} = \sqrt{5} \]. Thus, the foci are at \( (0, \pm\sqrt{5}) \) in the \( xy \)-plane.

Key concepts

Ellipse Foci

In geometry, particularly when dealing with ellipses, the foci (plural of focus) are two significant points. These points lie along the major axis of the ellipse. The ellipse itself is defined such that the sum of the distances from any point on the ellipse to each focus is constant, which is fundamental in describing its shape.

For example, consider the ellipse described by the equation \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \). The distance from the center to each focus, denoted by \( c \), is calculated using \( c = \sqrt{a^2 - b^2} \). This is a crucial property, as it determines the positions of the foci, often aiding in visualizing and solving problems related to ellipses.

For the ellipse formed by the intersection given in the exercise, \( a = 3 \) and \( b = 2 \). So, the distance to each focus is \( c = \sqrt{3^2 - 2^2} = \sqrt{5} \). Thus, the coordinates of the foci are located at \((0, \pm\sqrt{5})\) along the y-axis.

Equation of Ellipse

The equation of an ellipse is paramount in defining its geometry on a coordinate plane. In its standard form, the equation is written as \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \( a \) and \( b \) correspond to the semi-major and semi-minor axes, respectively.

This form reveals several characteristics:

• The center of the ellipse is at the origin (0,0) if there are no linear terms.
• The ellipse is aligned with the coordinate axes unless rotation terms are present.
• The relative sizes of \( a^2 \) and \( b^2 \) determine its orientation and the lengths of its axes.

For the ellipse from the original exercise, setting \( z = 4 \) in the given surface equation \( z = \frac{x^2}{4} + \frac{y^2}{9} \) results in a new equation \( 1 = \frac{x^2}{4} + \frac{y^2}{9} \) in standard form. This shows the ellipse centered at the origin with its major axis parallel to the y-axis.

Semi-major and Semi-minor Axes

These terms describe half of the longest and shortest diameters of the ellipse, respectively. The semi-major axis is the longer of the two and denotes the direction along which an ellipse extends the furthest from its center. Conversely, the semi-minor axis is the shorter radius.

For an ellipse defined by \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), the semi-major axis has length \( a \), and the direction along this axis dictates the orientation of the ellipse:

• If \( a > b \): the major axis is on the \( y \)-axis.
• If \( b > a \): the major axis is on the \( x \)-axis.

In the given problem, after simplification, the resulting ellipse equation \( 1 = \frac{x^2}{4} + \frac{y^2}{9} \) highlights the semi-major axis (length 3) aligned with the y-axis, and the semi-minor axis (length 2) aligned with the x-axis.

Intersection of Geometric Shapes

Finding the intersection between geometric shapes like ellipses and planes is a common problem in calculus and analytic geometry. This involves determining where two or more surfaces or lines meet in space.

In this context, the problem features an elliptical cylinder described by \( z = \frac{x^2}{4} + \frac{y^2}{9} \) intersecting a horizontal plane \( z = 4 \). Steps to find such intersections:

• Replace the variable \( z \) in the elliptical cylinder's equation with the constant plane value.
• Simplify the resulting expression into an equation representing the intersection curve in the appropriate coordinate plane.

The outcome is an ellipse, given by \( 1 = \frac{x^2}{4} + \frac{y^2}{9} \). This illustrates how intersections yield new shapes, useful in 3D modeling and applications ranging from architecture to aerodynamics.