Solution - Episode 1: Conic Section

A conic section is defined by the equation 25x² + 4y² - 100x + 8y + 4 = 0

Answer the following given the equation above

a) What is the conic section?
B) write the equation in standard form
c) sketch the graph

Now to solve:

To solve these question first thing that i would do is question "B", because you need the standard form of the equation to know what it is and also sketch the graph.

B) Write the equation in standard form.

The First thing you should do is collect like terms so it should look like this.
25x² - 100x + 4y² + 8y = -4

The next would be to factor out 25 out of the x values it should look like this.
25(x² - 4x) + 4y² + 8y = - 4

The next would be to factor out 4 out of the y values it should look like this.
25(x² - 4x) + 4(y² + 2y) = - 4

Now we have to complete the squares. (AND DON'T FORGET IF YOU ADD ON ONE SIDE YOU HAVE TO ADD IT ON THE OTHER SIDE TOO AND ALSO DON'T FORGET TO MULTIPLY OUT WHAT YOU ADD. Ex. 25(x² - 4x + 4) the four is actually 100.)

25(x² - 4x + 4) + 4(y² + 2y + 1) = - 4 + 4 + 100

After you have done that you have to multiply all the terms by the reciprocals of the coefficients so it should look like this.



And there you go now have put that equation into standard form.


Now for Question A)

A) What is the conic section?

Well the answer is that it is an ellipse. You can notice that it is an ellipse because the denominators are different.

NOW to graph C)

C) Sketch the graph

TO sketch the graph you first find the center which in this case it is (2, -1)
Next you would check with denominator is larger so you can determine if it is a vertical or horizontal ellipse. Well this ellipse is a vertical ellipse.
after you find the semi-Transverse axis and the semi-minor axis. to find that you need to take the square roots of each denominator. Which are 2 and 5.
After you have that you start from the center and go 2 units from to the left and 2 unit to the right and put points at those spots. That should give you the minor axis.
After you have that you start from the center and go 5 units up and 5 unit down and put points at those spots . That should give you your transverse axis.
Now that you have those points just connect the dots and Ta-da you have sketched the ellipse

Should look something like this