How to find the equations of the axis of the ellipse

In summary, the conversation discusses finding the equations of the axis of an ellipse given a specific equation. The method involves using the concept of a "middle line" and finding the directions of the axes, which can be calculated using a general formula. The equations of the axes can then be found using the partial derivatives of the function.
  • #1
pbialos
I was hoping you could give me a hint on how to find the equations of the axis of the ellipse of equation [tex]5x^2-6xy+5y^2-4x-4y-4=0[/tex]. I think this is supposed to be an exercise about lagrange multipliers or something related to the gradient, but i really don't know. I am clueless, i don't know any property about ellipses in general.

Many Thanks, Paul.
 
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  • #2
Unfortunately, I don't know the English terminology that well regarding analytic geometry.

There is a thing which we call "middle line", which is a polar (polar line) of a point at infinity with respect to the conic (here the ellipse).

Axes are a special case of these lines, two of those whose directions are perpendicular, so where [itex]m_1 = - \frac{1}{{m_2 }}[/itex], where m is a direction.

For a general conic [itex]ax^2 + 2b''xy + a'y + 2by + 2b'x + a'' = 0[/itex] those direction are the solutions of [itex]b''m^2 + \left( {a - a'} \right)m - b'' = 0[/itex].

In this case, you get [itex]m = 1\,\,\, \vee \,\,m = - 1[/itex].

Now, the equations of the axes are then:

[tex]\begin{array}{l}
F_x ^\prime \left( {x,y} \right) + m_1 \cdot F_y ^\prime \left( {x,y} \right) = 0 \\
F_x ^\prime \left( {x,y} \right) + m_2 \cdot F_y ^\prime \left( {x,y} \right) = 0 \\
\end{array}[/tex]

Here, [itex]F_x ^\prime \left( {x,y} \right)[/itex] mean the partial derivative of the function to the variable x (same for y).

I tried it and it seems to be working, can you get the equations now?
 
  • #3


Hi Paul,

To find the equations of the axes of an ellipse, you can start by rewriting the given equation in standard form, which is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively.

To do this, you can use the method of completing the square to group the terms with x and y together. Once you have the equation in standard form, you can easily identify the center and lengths of the axes.

In this case, the equation can be rewritten as (5x^2-4x) + (-6xy+5y^2-4y) = 4. Then, completing the square for the terms with x and y, we get (5(x^2-4/5x)) + (-6(x^2-4/5xy)) + (5(y^2-4/5y)) = 4. Simplifying and factoring, we get (5(x-2/5)^2) + (-6(x-2/5)(y-2/3)) + (5(y-2/3)^2) = 4.

Now, comparing this to the standard form equation, we can see that the center is at (2/5, 2/3) and the lengths of the axes are (5/√4) and (5/√5). Therefore, the equations of the axes are x-2/5 = 0 and y-2/3 = 0.

I hope this helps! Remember to always look for patterns and use the properties of ellipses when solving these types of problems.

 

1. What is an ellipse?

An ellipse is a geometric shape that resembles a flattened circle. It is defined as the set of all points in a plane whose distances from two fixed points, called foci, have a constant sum.

2. What are the equations of the axis of an ellipse?

The equations of the axis of an ellipse are x = h ± a and y = k ± b, where (h,k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis. The plus sign corresponds to the axis passing through the foci, and the minus sign corresponds to the axis passing through the vertices.

3. How do I find the center of an ellipse?

The center of an ellipse can be found by using the formula (h,k), where h is the x-coordinate of the center and k is the y-coordinate of the center. These values can be determined by finding the midpoint between the foci or by using the endpoints of the major and minor axes.

4. What is the difference between the semi-major and semi-minor axes of an ellipse?

The semi-major axis is the longest distance from the center of an ellipse to the edge, while the semi-minor axis is the shortest distance from the center to the edge. The semi-major axis is always perpendicular to the semi-minor axis and their lengths are related by the equation a²=b²+c², where a is the length of the semi-major axis, b is the length of the semi-minor axis, and c is the distance between the center and one of the foci.

5. How do I know if an ellipse is horizontal or vertical?

An ellipse is horizontal if the major axis is parallel to the x-axis and vertical if the major axis is parallel to the y-axis. This can be determined by looking at the length of the semi-major and semi-minor axes. If the length of the semi-major axis is greater than the length of the semi-minor axis, the ellipse is horizontal. If the length of the semi-minor axis is greater than the length of the semi-major axis, the ellipse is vertical.

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