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hyperbolas orthogonal to ellipses
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(Example)
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Let $a^2 > b^2$ , $s > -b^2$ and $a^2 > t > b^2$ . Show that each of the ellipses
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(1) |
is an orthogonal curve of every hyperbola
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(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: $$\frac{x_0x}{a^2+s}+\frac{y_0y}{b^2+s} = 1, \quad
\frac{x_0x}{a^2-t}-\frac{y_0y}{t-b^2} = 1$$ Solving these equations for $y$ shows that the slopes of the tangents are $$m_1 = -\frac{b^2+s}{a^2+s}\!\cdot\!\frac{x_0}{y_0}, \quad m_2 = -\frac{t-b^2}{a^2-t}\!\cdot\!\frac{x_0}{y_0},$$ and thus their product is
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(3) |
On the other hand, the point $(x_0,\,y_0)$ satisfies the equation gotten from (1) and (2) via subtraction: $$0 = \left(\frac{1}{a^2+s}-\frac{1}{a^2-t}\right)x_0^2+\left(\frac{1}{b^2+s}-\frac{1}{t-b^2}\right)y_0^2 = (s+t)\left(\frac{-x_0^2}{(a^2+s)(a^2-t)}+\frac{y_0^2}{(b^2+s)(t-b^2)}\right)$$ Since $s\!+\!t$ cannot be 0, the second factor of the abobe product must vanish, which implies the proportion equation $$\frac{x_0^2}{(a^2+s)(a^2-t)} = \frac{y_0^2}{(b^2+s)(t-b^2)}.$$ Utilising this in the equation (3) yields the condition of orthogonality $$m_1m_2 = -1,$$ for the tangents, which means that the ellipse and the hyperbola intersect orthogonally.
Note. Both the ellipses (1) and the hyperbolas (2) have the common foci $(\pm\sqrt{a^2\!-\!b^2},\,0)$ , being thus confocal.
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"hyperbolas orthogonal to ellipses" is owned by pahio. [ full author list (2) ]
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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.
Accessed 1294 times total.
Classification:
| AMS MSC: | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) |
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Pending Errata and Addenda
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