Write equation of translated graph in general form

[wvcaudill2]

Homework Statement

Identify the graph of each equation. Write the equation of the translated graph for T_(-5,6) in general form. Then draw the graph.

4x² + 5y² = 20

Homework Equations

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The Attempt at a Solution

I put the equation instandard form and got :

\frac{x^2}{5} + \frac{y^2}{4} = 1

This shows that it is a graph of an ellipse with center (0,0).

However, I wasnt sure what to do next. I found the translated points using matrices and adding, but I don't know what to do with this, ot even if I need this information.

 

[Gear.0]

Is it standard form?
let's see,
(standard form) = 0
and your equation is
(some equation) = 1

 

[Mark44]

Homework Statement

Identify the graph of each equation. Write the equation of the translated graph for T_(-5,6) in general form. Then draw the graph.

4x² + 5y² = 20

Homework Equations

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The Attempt at a Solution

I put the equation instandard form and got :

\frac{x^2}{5} + \frac{y^2}{4} = 1

This shows that it is a graph of an ellipse with center (0,0).

However, I wasnt sure what to do next. I found the translated points using matrices and adding, but I don't know what to do with this, ot even if I need this information.

If the graph of the ellipse is translated left by 5 units and up 6 units, the center will move from (0, 0) to (-5, 6). How will this affect the equation
\frac{x^2}{5} + \frac{y^2}{4} = 1?

Tip: write your entire equation in one pair of [ tex] tags.

 

[wvcaudill2]

If the graph of the ellipse is translated left by 5 units and up 6 units, the center will move from (0, 0) to (-5, 6). How will this affect the equation
\frac{x^2}{5} + \frac{y^2}{4} = 1?

Tip: write your entire equation in one pair of [ tex] tags.

The equation will then become:

\frac{(x+5)^2}{5} + \frac{(y-6)^2}{4} = 1

NOw, how can I convert this into general form?

 

[Mark44]

Multiply both sides by 20, expand (x + 5) and (y - 6)², and bring everything to one side, with zero on the other side.

 

[wvcaudill2]

Multiply both sides by 20, expand (x + 5) and (y - 6)², and bring everything to one side, with zero on the other side.

Ok, I get the following:


4x^2 + 5y^2 + 40x - 60y +260 = 0

This looks good, except there is no "Bxy" term. When would this term appear?

Looking through my older notes, I found a few problems where i converted from standard form to general form before. However, it looks like that way I did it was by setting up a system of 3 equations, and somehow solving for the variables D, E, and F. Can this method I mentioned by used on anything other than circles?


Lastly, will the method you showed me work with all types of conic sections?

 

[Mark44]

The xy term shows up when conic sections (such as an ellipse in this problem) are rotated.

Looking through my older notes, I found a few problems where i converted from standard form to general form before. However, it looks like that way I did it was by setting up a system of 3 equations, and somehow solving for the variables D, E, and F. Can this method I mentioned by used on anything other than circles?

I don't understand what you're saying here. To convert from general form to standard form (if I remember these terms correctly), you complete the square in the x terms and in the y terms. If you're talking about the general form with an xy term, it's been a very long time since I've done that, so can't commit on the method you describe.

 

[HallsofIvy]

If you have a conic section with Ax^2+ Bxy+ Cy^2 (and other non-squared terms) with C not 0, you can get rid of it in either of two ways:

1) Let x= x' cos(\theta)- y' sin(\theta), y= x' sin(\theta)+ y' cos(\theta), replace x and y in your formula by that, and solve for a value of \theta that make Bxy= 0. \theta is the angle through which the axes are rotated and the entire equation with x', y' instead of x, y will have no "xy" term.


2) Find the eigenvalues and eigenvectors of the matrix
\begin{bmatrix}A & \frac{1}{2}C \\ \frac{1}{2}C & D\end{bmatrix}

The eigenvectors will point along line y= ax and y= bx, that are parallel to the axes of the conic and substituting x'= ax- y, y'= bx- y will get rid of the "xy" term.