Problem 299 - Calculus Volume 3

Question

Sketch the graph of each conic. $$ \frac{x^{2}}{4}+\frac{y^{2}}{16}=1 $$

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin with vertices at (0, 4) and (0, -4), and co-vertices at (2, 0) and (-2, 0).

Step 1: Identify the Type of Conic

The given equation is \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \). This equation represents an ellipse because it is in the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where both \( x^2 \) and \( y^2 \) terms are positive and the coefficients are different.

Step 2: Find the Semi-major and Semi-minor Axes

In an ellipse of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), if \( a^2 < b^2 \), then \( b \) is the semi-major axis and \( a \) is the semi-minor axis. Here, \( a^2 = 4 \) (\( a = 2 \)) and \( b^2 = 16 \) (\( b = 4 \)). Thus, the semi-major axis is 4 (along the y-axis) and the semi-minor axis is 2 (along the x-axis).

Step 3: Determine the Vertices

The vertices of the ellipse are found using the semi-major axis length along the y-axis. This gives the vertices at \((0, 4)\) and \((0, -4)\).

Step 4: Determine the Co-vertices

The co-vertices of the ellipse are found using the semi-minor axis length along the x-axis. This gives the co-vertices at \((2, 0)\) and \((-2, 0)\).

Step 5: Sketch the Ellipse

To sketch the ellipse, plot the vertices \((0, 4)\) and \((0, -4)\), and the co-vertices \((2, 0)\) and \((-2, 0)\) on a coordinate plane. Draw a smooth curve connecting these points to form an elongated circle, ensuring it is wider in the vertical direction (y-axis) than in the horizontal direction (x-axis).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. This intersection can result in different types of curves: circles, ellipses, parabolas, and hyperbolas. Each of these curves has unique characteristics and equations that describe their geometric properties.
Circle, ellipse, parabola, and hyperbola each have specific forms based on their intercept with the cone and the angle of the intersecting plane. Understanding these sections is crucial in geometry and algebra as they frequently appear in physics, engineering, and astronomy problems.
For our exercise, the equation represents an ellipse. Recognizing this required an understanding of the equation's structure, characterized by squared terms and distinct coefficients for each axis. Identifying the type of conic section is the first step in solving such problems and setting the stage for further analysis.

Ellipse Equation

The ellipse is one of the conic sections represented by a quadratic equation of the form:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]where:

• is the length of the semi-minor axis.
• is the length of the semi-major axis.

In the given problem, the equation \( \frac{x^{2}}{4} + \frac{y^{2}}{16} = 1 \) describes an ellipse. This standard form lets us easily extract the lengths of the semi-major and semi-minor axes by comparing it to \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
The equality to 1 ensures that the points \( (x, y) \) on the curve have the same proportional relationship to the lengths of the axes, thereby defining the shape of the ellipse. The knowledge of whether \( a^2 < b^2 \) or \( a^2 > b^2 \) helps determine the direction of the major axis.

Semi-major Axis

In an ellipse, the semi-major axis is the longest diameter, running through the center and foci, and defining its maximum width. It is vital in describing the ellipse's size and orientation.
For our exercise, we determined that the semi-major axis corresponds to the length 4 units, as computed from the equation based on \( b = 4 \), where \( b^2 = 16 \), indicating it lies along the y-axis.
Thus, the major axis stretches vertically from the point (0, 4) to (0, -4). Recognizing which axis is major is essential for plotting the graph and understanding the geometric layout of the ellipse in relation to its coordinate axes.

Semi-minor Axis

The semi-minor axis of an ellipse is perpendicular to the semi-major axis and defines its minimum width. Despite being shorter, it plays a critical role in specifying the ellipse's shape and the proportion of its axes.
For the provided ellipse equation, the semi-minor axis has a length of 2 units, derived from \( a = 2 \), where \( a^2 = 4 \). Since it is aligned with the x-axis, the ellipse's profile is elongated more in the vertical direction.
The semi-minor axis extends horizontally from the point (2, 0) to (-2, 0). This measure is significant when sketching the ellipse, as it helps define the balanced, but stretched nature of the curve, ensuring an accurate depiction and understanding of this special conic section.