Analytic Geometry And Principles of Algebra

Cover Analytic Geometry And Principles of Algebra
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Show that an ellipse and hyperbola with common foci are orthogonal.
6. Show that the eccentricity of a hyperbola is equal to the secant of half the angle between the asymptotes.
7. Express the cosine of the angle between the asymptotes of a hyperbola in terms of its eccentricity.
8. Show that the tangents at the vertices of a hyperbola intersect the asymptotes at points on the circle about the center through the foci.
9. Show that the point of contact of a tangent to a hyperbola is the midpoint
... between its intersections with the asymptotes.
10. Show that the area of the triangle formed by the asymptotes and any tangent to a hyperbola is constant.
11. Show that the product of the distances from the center of a hyper- bola to the intersections of any tangent with the asymptotes is constant.
12. Show that the tangent to a hyperbola at any point bisects the angle between the focal radii of the point. ( /''^? . | 212 PLANE ANALYTIC GEOMETRY [X, 219 13. As the sum of the focal radii of every point of an ellipse is con- stant ( 204) and the normal bisects the angle between the focal radii ( 216), a sound wave issuing from one focus is reflected by the ellipse to the other focus.


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