Plane Algebraic Curves
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31. The points of contact of the tangents from C to the quartic lie on a conic through C.
[(fh + bg) xz + (gh + of) yz = Ax 2 + By 2 +2 (ch - 2fg) xy. ] •Ex. 32. The points of contact of the tangents from A lie on a conic through A, C and the intersections of BC with the quartic. Similarly for the tangents from B. t [bgxz + ghyz = By 1 + (ch — 2fg) xy ;• fhxz + afyz = Ax 2 + (ch — 2fg) xy.] Ex. 33. The conies of Ex. 31, 32 pass through the same four points.
Ex. 34. The tangents at C meet the qu...artic again in C 1 and C 2 , and C y C % meets the curve again at C\ and C 2 . Show that the points of contact of the tangents from C lie on a conic touching CC\ and CC\ at C\ and C'j and passing through A and B.
Show also that through the intersections of the conic U = with this conic a conic can be drawn touching CA and CB at A and B.
[(/F+gG + hH)f-H(hz 2 + cxy) = cU.] Ex. 35. Through two pairs of vertices of the quadrilateral formed by the bitangents can be drawn (i) a conic touching CA at A and CB at B ; (ii) a conic touching conic I at its intersections with AB ; (iii) two conies each touching CA at A and CB at B and each passing through the points of contact of two bitangents.
What to read after Plane Algebraic Curves?
You can find similar books in the "Read Also" column, or choose other free books by Hilton, Harold, B.1876- to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
31. The points of contact of the tangents from C to the quartic lie on a conic through C.
[(fh + bg) xz + (gh + of) yz = Ax 2 + By 2 +2 (ch - 2fg) xy. ] •Ex. 32. The points of contact of the tangents from A lie on a conic through A, C and the intersections of BC with the quartic. Similarly for the tangents from B. t [bgxz + ghyz = By 1 + (ch — 2fg) xy ;• fhxz + afyz = Ax 2 + (ch — 2fg) xy.] Ex. 33. The conies of Ex. 31, 32 pass through the same four points.
Ex. 34. The tangents at C meet the qu...artic again in C 1 and C 2 , and C y C % meets the curve again at C\ and C 2 . Show that the points of contact of the tangents from C lie on a conic touching CC\ and CC\ at C\ and C'j and passing through A and B.
Show also that through the intersections of the conic U = with this conic a conic can be drawn touching CA and CB at A and B.
[(/F+gG + hH)f-H(hz 2 + cxy) = cU.] Ex. 35. Through two pairs of vertices of the quadrilateral formed by the bitangents can be drawn (i) a conic touching CA at A and CB at B ; (ii) a conic touching conic I at its intersections with AB ; (iii) two conies each touching CA at A and CB at B and each passing through the points of contact of two bitangents.
What to read after Plane Algebraic Curves?
You can find similar books in the "Read Also" column, or choose other free books by Hilton, Harold, B.1876- to read onlineMoreLess
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