The Geometry of Conics, Part 1 |
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Page 34
... Auxiliary Circle . 72. PROP . V. If the ordinate NP of any point P on the ellipse be produced to meet the auxiliary circle in PN : pN = CB CA. : P , then D A R For and PN : AN . NA ' = CB : CA ' , AN . NA ' = pN2 . Therefore PN2 : ÎN2 ...
... Auxiliary Circle . 72. PROP . V. If the ordinate NP of any point P on the ellipse be produced to meet the auxiliary circle in PN : pN = CB CA. : P , then D A R For and PN : AN . NA ' = CB : CA ' , AN . NA ' = pN2 . Therefore PN2 : ÎN2 ...
Page 35
... auxiliary circle in p , d , the angle p Cd will be a right angle , and the triangles pCN , CdR will be equivalent , so that CR - pN , and CN = d . Hence PN : CR CB : CA = DR : CN . [ 72 . 76. PROP . VII . The sum of the squares of ...
... auxiliary circle in p , d , the angle p Cd will be a right angle , and the triangles pCN , CdR will be equivalent , so that CR - pN , and CN = d . Hence PN : CR CB : CA = DR : CN . [ 72 . 76. PROP . VII . The sum of the squares of ...
Page 38
... Auxiliary Point of P. When P lies on the ellipse , p lies on the auxi- liary circle . [ 73 . 82. If any number of points O , P , Q ... lie on a straight line which meets the axis in T , their auxiliary points o , p , q ... will lie on a ...
... Auxiliary Point of P. When P lies on the ellipse , p lies on the auxi- liary circle . [ 73 . 82. If any number of points O , P , Q ... lie on a straight line which meets the axis in T , their auxiliary points o , p , q ... will lie on a ...
Page 39
... auxiliary point o is fixed . There- fore op . oq is constant , whether o be without or within the auxiliary circle . [ Euc . III . 35 , 36 . Hence , the radius Cp being constant , the ratio of OP . OQ to CD2 is constant whether O be ...
... auxiliary point o is fixed . There- fore op . oq is constant , whether o be without or within the auxiliary circle . [ Euc . III . 35 , 36 . Hence , the radius Cp being constant , the ratio of OP . OQ to CD2 is constant whether O be ...
Page 40
... circle on BB ' is in fact a Minor Auxiliary Circle , having auxiliary properties corresponding to those of the circle on AA ' . 92. PROP . XIII . To find the length of 40 CHORD - PROPERTIES OF THE ELLIPSE .
... circle on BB ' is in fact a Minor Auxiliary Circle , having auxiliary properties corresponding to those of the circle on AA ' . 92. PROP . XIII . To find the length of 40 CHORD - PROPERTIES OF THE ELLIPSE .
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Common terms and phrases
abscissa auxiliary circle bisects the angle Bishop of Carlisle CA² Cambridge CB² CD² centre CN² concyclic Conic Sections conjugate diameters conjugate radii constant ratio CP² CS² CV² draw drawn ellipse equally inclined Fellow of Trinity fixed point focal chord focal perpendicular focal radii focus Geometry Hence intersect J. W. DONALDSON late Fellow latus rectum locus meet the asymptotes meet the curve meet the directrix meets the axis middle point minor axis normal parabola parallel chords parallelogram Pembroke College PN² point Q principal ordinate PROP QV² radius rectangular hyperbola right angles right-angled triangles Second Edition Shew shewn sides SP² St John's College straight line joining subtends a right supplementary angles tangent Third Edition Treatise Trinity College vertex
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