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Conic Sections, quadratic Equations, X,Y and U,V axis

Posted 10 years ago

I was given the following question on class and asked to solve the questions as followed. Could anybody help me out. I feel as though there is a typo within the equation. I don't understand how there is a subscript of 354y on the 72x term. Consider the conic section given by the quadratic equation

36x^2-24xy+29y^2+72x_354y+945=0. 1.Use the discriminant to tell whether this is an ellipse, a parabola or an hyperbola. 2.Find the angle or find the sin and cos of the angle through which the xy axes should be rotated so that, in the new uv axis system, there is no cross product term. 3.Write the quadratic equation in terms of u and v. 4.Write the equation in the standard form of whatever conic section you have found it to be. 5.Sketch the conic section showing both the xy and the uv axes. Label any relevant points and/or lines in the graph in terms of x and y.

POSTED BY: Ethin Kiekhafer
2 Replies

Here's a start on solution to your problem. Equation should probably read

36 x^2 - 24 x y + 29 y^2 + 72 x - 354 y + 945 = 0

Discriminant D = B^2 - 4 A C

In[16]:= (-24)^2 - 4*36*29

Out[16]= -3600

D < 0 so you have an ellipse

Now plot equation

In[22]:= Solve[36 x^2 - 24 x y + 29 y^2 + 72 x - 354 y + 945 == 0, y]

Out[22]= {{y -> 3/29 (59 + 4 x - 2 Sqrt[109 + 60 x - 25 x^2])}, {y -> 3/29 (59 + 4 x + 2 Sqrt[109 + 60 x - 25 x^2])}}

In[23]:= Plot[{3/29 (59 + 4 x - 2 Sqrt[109 + 60 x - 25 x^2]), 3/29 (59 + 4 x + 2 Sqrt[109 + 60 x - 25 x^2])}, {x, -10, 10}, PlotRange -> All]

Can't show output.

To put into standard form, complete the squares for x and y terms. Then you can figure out the angle of rotation.

POSTED BY: S M Blinder
Posted 10 years ago

The _ should be either a + 0r - sign, most likely a - . And items 1 through 5 look like a fun learning experience!

POSTED BY: David Keith
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