Question

For the following exercises, sketch the graph of each conic. $$ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$

Short answer

The graph is a horizontal ellipse centered at the origin with vertices at (±3, 0) and co-vertices at (0, ±2).

Step-by-step solution

Identify the Type of Conic

The given equation is \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \). We see that both \( x^2 \) and \( y^2 \) are positive and the sum is equal to 1. This indicates that the conic is an ellipse.

Standard Form of the Ellipse

Since the equation is \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), it is already in the standard form of an ellipse, which is \( \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1 \). Identify \( a^2 \) and \( b^2 \): here \( a^2 = 9 \) and \( b^2 = 4 \).

Determine the Orientation

Compare \( a^2 \) and \( b^2 \) to determine orientation. Since \( a^2 > b^2 \), the ellipse is horizontally oriented.

Find the Center, Vertices, and Co-vertices

The center of the ellipse is at the origin \((0, 0)\). For a horizontal ellipse, the vertices are \( (\pm a, 0) \) and the co-vertices are \( (0, \pm b) \). Thus, \( a = 3 \) and \( b = 2 \). The vertices are \( (3, 0) \) and \( (-3, 0) \), and the co-vertices are \( (0, 2) \) and \( (0, -2) \).

Sketch the Ellipse

Plot the center of the ellipse at (0, 0). Mark the vertices at (3, 0) and (-3, 0), and the co-vertices at (0, 2) and (0, -2). Draw a smooth, oval-shaped curve connecting these points to form the ellipse.

Key concepts

Ellipse

An ellipse is a type of conic section that you encounter in analytic geometry. It's the shape you get when you take a double cone and slice it at an angle that is not perpendicular to its base. To visualize an ellipse, imagine a circle that is stretched either horizontally or vertically, giving it its distinct oval shape. In mathematical terms, an ellipse can be defined by its equation in standard form:

• For a horizontally stretched ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a > b \).
• For a vertically stretched ellipse: \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) where \( a > b \).

Key features of ellipses include:

• The center, which is the midpoint of the largest and smallest widths of the ellipse.
• The major axis, which is the longest diameter of the ellipse.
• The minor axis, perpendicular to the major axis, which is the shortest diameter of the ellipse.
• The semi-major axis \( a \) and the semi-minor axis \( b \), which are halves of the major and minor axes, respectively.

Understanding these concepts makes it easier to identify and work with ellipses in various mathematical problems.

Graph Sketching

Sketching the graph of an equation can be a delightful process, especially when it involves conic sections like ellipses. The goal is to understand the shape and key features of the conic, such as its center, vertices, and orientation.

When sketching an ellipse:

• First, identify the center of the ellipse from the equation. For the equation \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), the center is at the origin \((0,0)\).
• Next, determine the distances for the vertices and co-vertices. In this example, \( a^2 = 9 \) leads to \( a = 3 \), and \( b^2 = 4 \) results in \( b = 2 \).
• Since \( a > b \), the ellipse is horizontal. Vertices are then \((3,0)\) and \((-3,0)\), while co-vertices are \((0,2)\) and \((0,-2)\).
• Plot these points on a coordinate plane and draw a smooth curve connecting them to form the ellipse.

Graph sketching enhances visual understanding and is crucial for confirming algebraic solutions.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, merges algebra and geometry to describe geometric shapes in a coordinate plane. It provides a bridge between algebraic expressions and geometric figures, allowing for a powerful approach to analyze shapes like ellipses.

In analytic geometry, all geometrical shapes are represented using equations, and you can analyze their properties by manipulating these equations. For instance:

• The equation of a circle is \( x^2 + y^2 = r^2 \), where \( r \) is the radius.
• An ellipse is identified by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• Each conic section has a standard form equation which reveals specific characteristics like orientation and dimensions.

Analytic geometry helps in dealing with complex problems efficiently by providing tools such as graphical representation and algebraic manipulation, aiding in deeper insights into geometrical relationships.