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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
B2 - 4AC = 0, then it is a Parabola
B2 - 4AC < 0, then it is an Ellipse
B2 - 4AC > 0, then it is a Hyperbola
Now, let's identify the type of conic section for the given equation above as follows
B2 - 4AC = (192)2 - 4(153)(97)
= 36864 - 59364
= - 22500
Since B2 - 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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| Photo by Math Principles in Everyday Life |
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
Substitute the values of Sin θ and Cos θ to the two equations above
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 a2 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
If
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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| Photo by Math Principles in Everyday Life |