__Category__: Analytic Geometry, Algebra"Published in Suisun City, California, USA"

Sketch the graph for the conic section

__Solution__:

The given equation above represents a conic section. The general equation of any conic sections is given by the equation

To identify the type of a conic section, let's consider the following conditions, if

A = C and B = 0, then it is a Circle

B

^{2}- 4AC = 0, then it is a Parabola

B

^{2}- 4AC < 0, then it is an Ellipse

B

^{2}- 4AC > 0, then it is a Hyperbola

Now, let's identify the type of conic section for the given equation above as follows

B

^{2}- 4AC = (192)

^{2}- 4(153)(97)

= 36864 - 59364

= - 22500

Since B

^{2}- 4AC < 0, then the given equation is an Ellipse.

Since the equation of an Ellipse has an xy term, then the major and minor axes of an Ellipse are not parallel to x-axis and y-axis. First, let's get the angle of inclination of an Ellipse as follows

Next, get the value of Cos 2θ as follows

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Use the Half-Angle Formula to get the values of Sin θ and Cos θ as follows

Next, we need to simplify the given equation in order to eliminate the xy term. The equations that we will use are the following

and

Substitute the values of Sin θ and Cos θ to the two equations above

and

Substitute the two equations above to the given equation, we have

Divide both sides of the equation by 225, we have

Since the value of a

^{2}is at y'

^{2}, then the major axis is parallel to y' axis.

To draw the x' axis, we will use the values of Sin θ and Cos θ. Count 4 units to the right from the origin since the center of an Ellipse is C(0, 0) and then 3 units upward. Connect that point with the origin and we have now x' axis.

To draw the y' axis, we will use the values of Sin θ and Cos θ. Count 4 units upward from the origin since the center of an Ellipse is C(0, 0) and then 3 units to the left. Connect that point with the origin and we have now y' axis.

In the next procedure, we will use the x' axis and y' axis to sketch the graph of an Ellipse.

If

then

If

then

The distance of the two foci from the center of an ellipse, c is calculated as follows

The distance of the four end points of latera recta from the two foci of an ellipse, p is calculated as follows

Label the center, four vertices, two foci, and the four ends of the latera recta using the x' axis and y' axis. The graph of an Ellipse will be like this

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