An Elementary Treatise On Conic Sections
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Therefore the angle KSP is a right angle.
(3) If chords of a conic subtend a constant angle at a focus, the tangents at the ends of the chord loill meet on a fixed conic, and the chord will touch another fixed conic.
Let 2^ be the angle the chord subtends at the focus. Let a- ^ and a+^ be the vectorial angles of the extremities of the chord.
174 POLAR EQUATION OF A CONIC.
The equatiou of the chord ^vill be - = e cos ^ + sec /3 cos (6' - a), T __ I cos 3 or ^ = gcos^. cos ^+cos [9 -a) (i).
But (...i) is the equatiou of the tangeut, at the poiut whose vectorial angle is a, to the conic whose equation is IcosB , ^ = l + ecosj8. cos 6 (u).
Hence the chord always touches a fixed conic, whose eccentricity is e cos ^, and semi-latus rectum Z cos /3.
The equations of the tangents at the ends of the chord will be I - — e cos ^ + cos (^-a+/3), and -=gcos ^ + cos (^ — a-^).
Both these lines meet the conic Z - = g cos 6 + cos iS r -, I m the same point, viz. where 6 = a and - = e cos a + cos/3.
Hence, the locus of the intersection of the tangents at the ends of the chord is the conic Z sec Q = l-re sec B .
What to read after An Elementary Treatise On Conic Sections?
You can find similar books in the "Read Also" column, or choose other free books by Smith, Charles, 1844-1916 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Therefore the angle KSP is a right angle.
(3) If chords of a conic subtend a constant angle at a focus, the tangents at the ends of the chord loill meet on a fixed conic, and the chord will touch another fixed conic.
Let 2^ be the angle the chord subtends at the focus. Let a- ^ and a+^ be the vectorial angles of the extremities of the chord.
174 POLAR EQUATION OF A CONIC.
The equatiou of the chord ^vill be - = e cos ^ + sec /3 cos (6' - a), T __ I cos 3 or ^ = gcos^. cos ^+cos [9 -a) (i).
But (...i) is the equatiou of the tangeut, at the poiut whose vectorial angle is a, to the conic whose equation is IcosB , ^ = l + ecosj8. cos 6 (u).
Hence the chord always touches a fixed conic, whose eccentricity is e cos ^, and semi-latus rectum Z cos /3.
The equations of the tangents at the ends of the chord will be I - — e cos ^ + cos (^-a+/3), and -=gcos ^ + cos (^ — a-^).
Both these lines meet the conic Z - = g cos 6 + cos iS r -, I m the same point, viz. where 6 = a and - = e cos a + cos/3.
Hence, the locus of the intersection of the tangents at the ends of the chord is the conic Z sec Q = l-re sec B .
What to read after An Elementary Treatise On Conic Sections?
You can find similar books in the "Read Also" column, or choose other free books by Smith, Charles, 1844-1916 to read onlineMoreLess
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