Standard Form of 3x^2 - 4xy = 2 & Variable Change

This represents a hyperbola in the new variables x1 and y1. In summary, the equation 3x^2 - 4xy = 2 can be rewritten in standard form using new variables x1 and y1 as \frac{1}{5}(x_{1}^{2}-4x_{1}y_{1}+2y_{1}^{2}) =2, representing a hyperbola. The change of variables can be found using the equations \cos\theta = \frac{a+c-d}{\sqrt{b^2 + (a+c-d)^2}} and \sin\theta = \frac{b}{\sqrt{b^2 + (a+c-d)^
  • #1
stunner5000pt
1,461
2
Write the equation in terms of new caraibles so that it is in standard position and identify the curve

[tex] 3x^2 - 4xy = 2 [/tex]

here a = 3, b = -4, c = 0 , [tex] d = \sqrt{(-4)^2+(3-0)^2} = 5 [/itex]

[tex] \cos\theta = \frac{a+c-d}{\sqrt{b^2 + (a+c-d)^2}} = \frac{-2}{2\sqrt{5}} [/tex]
[tex] \sin\theta = \frac{b}{\sqrt{b^2 + (a+c-d)^2}} = \frac{4}{\sqrt{20}} [/tex]

so [tex] P = \frac{1}{\sqrt{5}} \left(\begin{array}{cc} -1&-2 \\ 2&-1 \end{array}\right) [/tex]

from X = PY i get
[tex] x = \frac{-1}{\sqrt{5}} (x_{1}-2y_{1}) [/tex]
[tex] y = \frac{-1}{\sqrt{5}} (2x_{1}+y_{1}) [/tex]
where x1 and y1 are the new variables
is this fine??

is this how you get the change of variables??
 
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  • #2
Yes, this is how you can get the change of variables. The equation in standard form is now: \frac{1}{5}(x_{1}^{2}-4x_{1}y_{1}+2y_{1}^{2}) =2
 

1. What is the standard form of a quadratic equation?

The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are constants and x is the variable.

2. What is the purpose of using standard form in equations?

The standard form of an equation allows us to easily compare and analyze different quadratic equations. It also helps us to identify important characteristics such as the vertex, intercepts, and symmetry of the graph.

3. How do you convert an equation into standard form?

To convert an equation into standard form, we need to make sure that all terms are on one side of the equation and the other side is equal to zero. We can do this by rearranging the terms and simplifying if necessary.

4. What is the role of variable change in an equation?

Variable change allows us to manipulate an equation to make it easier to solve or analyze. It involves replacing a variable with a new variable or expression that makes the equation more manageable.

5. How can you determine the nature of the graph of an equation in standard form?

The nature of the graph can be determined by analyzing the value of the coefficient a in the standard form equation. If a is positive, the graph opens upwards and has a minimum value. If a is negative, the graph opens downwards and has a maximum value. The value of b determines the x-coordinate of the vertex and the value of c determines the y-coordinate of the vertex.

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