Let us cut the ellipse

with the family of parallel lines

having the common slope $m$; the number $k$ being a parameter.  Substituting the expression (1) of $y$ to the equation of the ellipse, we get the quadratic equation

determining the abscissas of the intersection points $P_{1}$ and $P_{2}$.  The midpoint of the chord $P_{1}P_{2}$ of the ellipse is determined by the abscissa $x_{0}$, which is the arithmetic mean of the abscissas of $P_{1}$ and $P_{2}$, i.e. the roots of (2).  Using the well-known properties of quadratic equation, we get

The ordinate $y_{0}$ of the midpoint of the chord satisfies

Eliminating the parameter $k$ from the two last equations gives us

This equation says that the point  $(x_{0},\,y_{0})$  is situated on the line  $y=-\frac{b^{2}}{a^{2}m}x$,  which passes through the origin, the centre of the ellipse. So we have the

Theorem 1.  The diameter of ellipse, i.e. the bisector of the parallel chords of an ellipse, is a line passing through the centre of the ellipse.

We see from the last equation, that the slope $m^{{\prime}}$ of the diameter which bisects the chords with the slope $m$ is  $m^{{\prime}}=-\frac{b^{2}}{a^{2}m}$;  accordingly one has the symmetric equation

between $m$ and $m^{{\prime}}$.  From it one can infer that, conversely, the diameter with the slope $m$ bisects all chords which have the slope $m^{{\prime}}$.  If we have two diameters of the ellipse, each one bisecting the chords parallel to the other, then these chords are conjugate diameters of each other.  Apparently, the major axis and the minor axis are a pair of conjugate diameters.

If the line (1) especially is a tangent of the ellipse, then the points $P_{1}$ and $P_{2}$ and the midpoint  $(x_{0},\,y_{0})$  coincide, and thus the equation (3) gives the slope of the tangent:

So we may write the equation of the tangent

of the ellipse. This can be simplified to the well-memorable form

where  $(x_{0},\,y_{0})$  is the tangency point on the ellipse. One has also the

Theorem 2.  The tangent of ellipse passing through an endpoint of a diameter is parallel to the conjugate diameter.

Notes.  The sum of squares of any pair of conjugate radii is equal to $a^{2}+b^{2}$.  The ellipse has only one pair of equally long conjugate diameters, viz. the ones lying on the diagonals of the rectangle  $x=\pm{a},\;y=\pm{b}$.  In the coordinate system  $(\xi,\,\eta)$  with coordinate axes along a pair of conjugate radii $\alpha,\,\beta$, the equation of the ellipse reads  $\frac{\xi^{2}}{\alpha^{2}}+\frac{\eta^{2}}{\beta^{2}}=1$.