Trilinear coordinates and other methods of modern analytical geometry of two dimensions; an elementary treatise

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1866 Excerpt: ...on the conic. The tangent at this point is df, 0df, df.. and the rectangle contained by the perpendiculars upon it from («„ fit, 7,), («,» & 78) is, (Art. 46) W df df""" dz". d/3" d7'J But by definition this is equal to a constant area (k, suppose), hence we obtain + (7A + 7A+3,cos5) Now this is a relation between the coordinates of any point whatever on the conic, and must therefore (accents suppressed) be the equation to the conic, and identical with the given equation which may be written (Art. 222) Hence the system of equations (1) may be Written «_ KK= U+Kas,-V+K0J3,-W-i-Ky.y, = 2W+K($lyt+/3,yl) m2V + K(yla, + y,l)--2 cos A--2 cos B-2 cos G (3; But instead of using all these equations which are somewhat complicated we may combine some of them with the simpler relations..,. 2A QQ 25' 2(7 expressing the fact that the centre bisects theL line joining the foci. (Art. 18. Cor.),/.' '".-.. Thus we have. _..-Ka1vt=2Act + Ka1t,-Kfifa-ilBfr + Kp, so that the equations (3) give V Kax + 2Aoti-U=KB? +25/8,-F= Ky + 2Cyx-W which with the identical relation connecting the coordinates of any point will be sufficient to determine the coordinates («„ fi» 7j There will generally be two imaginary solutions as welT as the two real ones, indicating two imaginary points having the property enunciated of the two real ones. Cob. If the conic be a parabola, K= 0 and the equations reduce to--2Aal-U= 25/5,-V= 2C7,-W. Obs. The equations in the form in which we have written them hold equally whether the coordinates be trilinear or v 275. To find equations to determine the vertices of the conic whose equation is Let («Vv8V 7') he the vertex; then the tangent? Hence if the coordinates be trilinear (Art. ...