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[parent] hyperbolas orthogonal to ellipses (Example)

Let $ a^2 > b^2$, $ s > -b^2$ and $ a^2 > t > b^2$. Show that each of the ellipses

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

is an orthogonal curve of every hyperbola
$\displaystyle \frac{x^2}{a^2-t}-\frac{y^2}{t-b^2} = 1.$ (2)

Let $ (x_0,\,y_0)$ be an intersection point of an ellipse (1) and a hyperbola (2). By polarizing both equations in the point $ (x_0,\,y_0)$ we get the equations of the tangents of the curves in this point:

$\displaystyle \frac{x_0x}{a^2+s}+\frac{y_0y}{b^2+s} = 1, \quad \frac{x_0x}{a^2-t}-\frac{y_0y}{t-b^2} = 1$
Solving these equations for $ y$ shows that the slopes of the tangents are
$\displaystyle m_1 = -\frac{b^2+s}{a^2+s}\!\cdot\!\frac{x_0}{y_0}, \quad m_2 = -\frac{t-b^2}{a^2-t}\!\cdot\!\frac{x_0}{y_0},$
and thus their product is
$\displaystyle m_1m_2 = -\frac{(b^2+s)(t-b^2)}{(a^2+s)(a^2-t)}\!\cdot\!\frac{x_0^2}{y_0^2}.$ (3)

On the other hand, the point $ (x_0,\,y_0)$ satisfies the equation gotten from (1) and (2) via subtraction:
$\displaystyle 0 = \left(\frac{1}{a^2+s}-\frac{1}{a^2-t}\right)x_0^2+\left(\frac... ... = (s+t)\left(\frac{-x_0^2}{(a^2+s)(a^2-t)}+\frac{y_0^2}{(b^2+s)(t-b^2)}\right)$
Since $ s\!+\!t$ cannot be 0, the second factor of the abobe product must vanish, which implies the proportion equation
$\displaystyle \frac{x_0^2}{(a^2+s)(a^2-t)} = \frac{y_0^2}{(b^2+s)(t-b^2)}.$
Utilising this in the equation (3) yields the condition of orthogonality
$\displaystyle m_1m_2 = -1,$
for the tangents, which means that the ellipse and the hyperbola intersect orthogonally.

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Note. Both the ellipses (1) and the hyperbolas (2) have the common foci $ (\pm\sqrt{a^2\!-\!b^2},\,0)$, being thus confocal.



"hyperbolas orthogonal to ellipses" is owned by pahio. [ full author list (2) ]
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See Also: orthogonal curves, zero rule of product, pencil

Other names:  ellipses orthogonal to hyperbolas
Keywords:  confocal ellipses, confocal hyperbolas

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Cross-references: confocal, foci, condition of orthogonality, proportion equation, implies, vanish, subtraction, product, tangents, slopes, tangents of the curve, equations, polarizing, hyperbola, ellipse, point, intersection, orthogonal curve
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This is version 8 of hyperbolas orthogonal to ellipses, born on 2008-06-15, modified 2008-09-18.
Object id is 10704, canonical name is HyperbolasOrthogonalToEllipses.
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Classification:
AMS MSC51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

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