Briot and Bouquet's Elements of Analytical Geometry of Two Dimensions

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. 1896 Excerpt: ...through the foci, is the equation of the ensemble of the axis of the conic 137, eq. (24). 2 The investigation of the foci is simplified if one form the condition (d) an2 + 2 b-i + cr-+ 2 da + 2 ew + f = 0, which expresses the condition that the straight line nx + vy + 1 = 0 is tangent to the given conic ( 126). Let (-, fi) be a focus, the axis being rectangular; the straight line y--f i--i(x--u) = 0 ought to be tangent to the curve. Whence, for this particular straight line one has i-1 u =----, v =. f i--ia f i--iu Expressing the condition that these values of u and v satisfy the condition (d), one has, on developing and replacing i2 %-1, (') f (a1-/F)-2 d- + 2 e/3 + a-c + 2 ' (tap-e--d/? + b). The straight line ij-f i + i(x-a)=0 being tangent to the conic, one gets a second condition which leads to the preceding by changing i into--i. Therefore, if the real or imaginary point (a, p) is a focus, the coefficient of i and the term independent of i ought to be zero separately in the condition (c), and one has the two equations f (-!-/3s)-2 ia + 2 ep + a-c = 0, tup-e--d/3 + b = 0, already attained above eq. (r) in another form. In case the axes are oblique and include an angle 6, one expresses the condition that the two straight lines y--p = (cos 6 i sin ff)(x--a) are tangent to the conic. Investigation Of Secants Common To Two Curves Of The Second Degree. 285. We have seen that two curves of the second degree, S = 0, Si = 0, have in general four points in common; through these four points, which we will suppose for the present distinct, one can pass three pairs of straight lines. In case the curves are real, the common points are real, or conjugate imaginaries taken by pairs. There are three cases to consider: ...