# How to find the equations of the axis of the ellipse

pbialos
I was hoping you could give me a hint on how to find the equations of the axis of the ellipse of equation $$5x^2-6xy+5y^2-4x-4y-4=0$$. I think this is supposed to be an exercise about lagrange multipliers or something related to the gradient, but i really dont know. I am clueless, i dont know any property about ellipses in general.

Many Thanks, Paul.

## Answers and Replies

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TD
Homework Helper
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 $m_1 = - \frac{1}{{m_2 }}$, where m is a direction.

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

In this case, you get $m = 1\,\,\, \vee \,\,m = - 1$.

Now, the equations of the axes are then:

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

Here, $F_x ^\prime \left( {x,y} \right)$ 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?