The Cone and Its Sections Treated Geometrically

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Hamilton, Adams & Company, 1875 - Conic sections - 148 pages
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Page 70 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz.
Page 100 - On opposite sides of any chord of a rectangular hyperbola are described equal segments of circles ; shew that the four points in which the circles to which these segments belong again meet the hyperbola are the angular points of a parallelogram.
Page 49 - Д0 be the area of the triangle formed by joining the points of contact...
Page 141 - Given a right cone and a point within it, there are but two sections which have this point for focus ; and the planes of these sections make equal angles with the straight line joining the given point and the vertex of the cone. 123. If the curve formed by the intersection...
Page 98 - ... conies, if we draw any tangent to one and tangents to the second where this line meets it, these tangents will intersect on the normal to the first conic.
Page 7 - We may observe that the asymptotes intersect this circle in the same points as the directrices. An important property is: the difference of the focal distances of any point on the curve equals the transverse axis. The tangent at any point bisects the angle between the focal distances of the point, and the normal is equally inclined to the focal distances.
Page 48 - The locus of the foot of the perpendicular from the focus on a moving tangent is the circle on the major axis as diameter. 3. The locus of the point of intersection of perpendicular tangents is a circle with radius Va>
Page 141 - ... and passing through/'. Two solutions are generally possible.] 3. Shew that all sections of a right cone, made by planes parallel to tangent planes of the cone, are parabolas, and that the foci lie on a cone having with the first a common vertex and axis. [Shew that the foci of parallel sections lie on a straight line through the vertex.] 4. Find the least angle of a cone from which it is possible to cut an hyperbola, whose eccentricity shall be the ratio of two to one.
Page 65 - M, N; prove that the area of the triangle CPM varies inversely as that of the triangle CPN.
Page 67 - Prove that the distance between the two points on the circumference of an ellipse, at which a given chord, not passing through the centre, subtends the greatest and least angles, is equal to the diameter which bisects that chord.

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