Question

For the following exercises, sketch the graph of each conic. $$ 4 x^{2}+9 y^{2}=36 $$

Short answer

The graph is an ellipse centered at the origin with semi-major axis 3 along the x-axis, and semi-minor axis 2 along the y-axis.

Step-by-step solution

Identify the Type of Conic Section

The equation given is \(4x^2 + 9y^2 = 36\). Compare it with the standard form for ellipses \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Both the \(x\) and \(y\) terms are squared and have positive coefficients, and the equation is in the form \(Ax^2 + By^2 = C\), indicating it represents an ellipse.

Convert to Standard Form

Divide the entire equation by 36 to transform it into standard form:\[ \frac{4x^2}{36} + \frac{9y^2}{36} = 1 \]\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] This gives the standard form \(\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1\). Hence, this is an ellipse centered at the origin with semi-major axis \(a = 3\) and semi-minor axis \(b = 2\).

Determine the Orientation

Since the larger denominator is under the \(x^2\) term, the major axis of the ellipse lies along the x-axis. This means the ellipse is horizontally oriented.

Sketch the Graph

To sketch the graph:1. Center the ellipse at the origin \((0,0)\).2. Plot the vertices along the major axis (x-axis) at \((3,0)\) and \((-3,0)\).3. Plot the co-vertices along the minor axis (y-axis) at \((0,2)\) and \((0,-2)\).4. Draw the ellipse through these points, ensuring it is longer along the x-axis.

Key concepts

Ellipse

An ellipse is one of the four types of conic sections, which include ellipses, circles, parabolas, and hyperbolas. It is formed by slicing a cone with a plane at an angle such that it does not intersect the base of the cone. An ellipse looks like an elongated circle and has a smooth, symmetric shape.

There are two axes in an ellipse: the major axis and the minor axis. The major axis is the longest line that can be drawn through the center of the ellipse, whereas the minor axis is the shortest. The points where these axes intersect the ellipse are called vertices for the major axis and co-vertices for the minor axis.

• The center of the ellipse is the midpoint of both the major and minor axes.
• A special property of ellipses is that the sum of the distances from any point on the ellipse to two fixed points, called foci, remains constant.

Understanding these basic properties can help in visualizing and sketching an ellipse.

Standard Form

The standard form of an ellipse is a simplified mathematical model of its equation, making it easier to identify key features like the axes' lengths and the orientation. The equation of an ellipse in standard form is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Here, \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

When the equation is rearranged to fit this form, crucial details about the ellipse's dimensions and placement become evident. By comparing the coefficients of the squared terms, one can easily determine which axis is the major or minor.

• If \(a > b\), the ellipse is oriented horizontally.
• If \(b > a\), it's oriented vertically.
• The equation must be equated to 1 to confirm it's in the standard form.

Transforming any elliptical equation into this standard form simplifies further analyses and graphing steps.

Graph Sketching

Graph sketching for conic sections, like ellipses, requires understanding both the equation and its geometric interpretation. By transforming the equation into standard form, we can easily plot the shape on a graph.

Start by identifying the center from the standard form equation. For an ellipse centered at the origin, this point is \((0,0)\). Then, identify the lengths of the semi-major and semi-minor axes, \(a\) and \(b\), respectively.

• Locate the vertices of the ellipse on the major axis at distances \(a\) from the center.
• Locate the co-vertices on the minor axis at distances \(b\) from the center.
• Connect these points smoothly, ensuring the shape is longer along the major axis.

Remember, when the ellipse is centered at the origin, these points directly dictate the basic shape and orientation of the graph.

Semi-Major Axis

The semi-major axis is a critical feature of any ellipse, denoting half the length of its longest diameter. In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(a\) represents the semi-major axis.

Knowing the semi-major axis helps in determining the orientation and size of the ellipse:

• The full length of the major axis is \(2a\).
• The endpoints of the major axis are the vertices of the ellipse, located at \((\pm a, 0)\) for a horizontally oriented ellipse.
• For vertically oriented ellipses, vertices are at \((0, \pm a)\).

Comprehending the semi-major axis's role is essential for accurately graphing and analyzing an ellipse's characteristics.