PlanetMath: conjugate diameters of ellipse

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conjugate diameters of ellipse

(Topic)

Let us cut the ellipse

$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$

with the family of parallel lines

$\displaystyle y = mx+k$

(1)

having the common slope

; the number

being a parameter. Substituting the expression (1) of

to the equation of the ellipse, we get the quadratic equation

$\displaystyle (a^2m^2+b^2)x^2+2a^2mkx+a^2(k^2-b^2) = 0$

(2)

determining the abscissas of the intersection points

and

. The midpoint of the chord

of the ellipse is determined by the abscissa

, which is the arithmetic mean of the abscissas of

and

, i.e. the roots of (2). Using the well-known properties of quadratic equation, we get

$\displaystyle x_0 = -\frac{2a^2mk}{a^2m^2\!+\!b^2}\!:\!2 = -\frac{a^2mk}{a^2m^2+b^2}.$

The ordinate

of the midpoint of the chord satisfies

$\displaystyle y_0 = mx_0+k.$

Eliminating the parameter

from the two last equations gives us

$\displaystyle y_0 = -\frac{b^2}{a^2m}x_0.$

(3)

This equation says that the point $(x_0,\,y_0)$ is situated on the line $y = -\frac{b^2}{a^2m}x$ , 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 of the diameter which bisects the chords with the slope is $m' = -\frac{b^2}{a^2m}$ ; accordingly one has the symmetric equation

$\displaystyle mm' = -\frac{b^2}{a^2}$

between

and

. From it one can infer that, conversely, the diameter with the slope

bisects all chords which have the slope

. 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 and and the midpoint $(x_0,\,y_0)$ coincide, and thus the equation (3) gives the slope of the tangent:

$\displaystyle m_t = -\frac{b^2x_0}{a^2y_0}$

So we may write the equation of the tangent

$\displaystyle y-y_0 = -\frac{b^2x_0}{a^2y_0}(x-x_0)$

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

$\displaystyle \frac{x_0x}{a^2}+\frac{y_0y}{b^2} = 1$

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.

$\begin{pspicture}(-4.2,-2.5)(4.5,2.5) \psaxes[Dx=9,Dy=9]{->}(0,0)(-4.2,-2.4)(4.5... ...097) \rput(0,-2.5){A pair of conjugate diameters (blue and red)} \end{pspicture}$

Notes. The sum of squares of any pair of conjugate radii is equal to . 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 $\frac{\xi^2}{\alpha^2}+\frac{\eta^2}{\beta^2} = 1$ .