Find the Trajectory of Points

Proposition 20 PROBLEM 12

Around a given focus to describe any trajectory given in specie which shall pass through given points, and touch right lines given by position.

CASE 1

Around the focus S it is required to describe a trajectory ABC, pass ing through two points B, C. Because the the ratio of the trajectory is given in specie, GAS principal axis to the distance of the foci to BS, and In that ratio take will be given. KB LC H CS. to About and on the the with the intervals BK, CL, describe two circles and in let fall K the that touches the same line L, KL, perpendicu right which cut in A and a, so that GA may be to AS, and Ga to aS, lar SG as KB to BS and with the axis A., and vertices A, a, describe a trajectory be the other focus of the described For let I say the thing is done. Ga to aS, then by division we shall GA to AS as and is seeing figure, have Ga GA, or Aa to S AS, or SH in the same ratio and therefore in the ratio which the principal axis of the figure to be described has to the distance of its foci and therefore the described figure is of the same And since KB to BS, species with the figure which was to be described. and LC to CS, are in the same ratio, this figure will pass through tht- points B, C, as is manifest from the conic sections.

CASE 2

About the focus S it is required to describe a trajectory which shall somewhere centres B, C,

touch two right lines on those tangents ST, may TR, From tr. the focus the perpendiculars St, which produce to V, v, so that TV, tv be equal to TS, tS. Bisect Vv in O, and fall let erect the indefinite perpendicular OH, and cut the right line VS infinitely produced in and K k, so that VK KS, and VA* be to j I. V to A~S, as the principal OH K/J describe a circle cutting H in ; axis of the tra On jectory to be described is to the distance of its foci. and with the the diameter foci S, I say, the thing principal axis equal to VH, describe a trajectory For bisecting in X, and joining HX, HS, HV, Hv, because : as VA- to AS ; and by composition, as by division, as VA 2KX VXH, to 2SX, and HXS VK kS to therefore as will be similar and therefore as done. VK Kk KS H, and is VK described trajectory, ; KS. VX KS, to VK that -f- is, HX and HX to therefore VH KS + kS 2VX to 2KX, V/c to as will be to is ; and the triangles to as SX, SH to and VX XH ; Wherefore VH, the principal axis of the has the same ratio to SH, the distance of the foci, as to12S

the principal axis of the trajectory which was to be described has to the distance of its foci and is therefore of the same species. Arid seeing VH, ; vH are equal to the principal axis, and VS, vS are perpendicularly bisected by the right lines TR, tr, it is evident (by Lem. XV) that those right lines touch the described trajectory.

CASE 3

About the focus S 3. TR shall touch a right line Jet fall the perpendicular ST it is required to describe a trajectory, which On the right line TR in a given Point R. ST, which produce to V, so that TV may be join VR, and cut the right line VS indefinitely produced in and k, so. that may be to SK, and V& to SAr, as the principal axis of the ellipsis to be described to the distance of its foci ; and on the equal to ; K VK diameter KA= describing a circle, H VR H cut the then with right line produced in the foci S, H, and principal axis equal to VH, describe a trajectory I say, the thing to SK, is to as is done. For ; R .+++ : VH VK SH K S and therefore as the principal axis of the trajectory which was to be de scribed to the distance of its foci (as appears from what we have demon strated in Case 2) ; and therefore the described trajectory species with that which was to be described is of the same but that the right line TR, is bisected, touches the VRS the which angle trajectory in the point R, by is certain from the properties of the conic sections.

CASE 4

About the focus S it ; r is a trajectory required APB that shall touch a right line describe to TR, and pass through any given point P without the tangent, and shall be similar to the figure apb, described with the principal axis ab, and foci s, h. On the tangent TR the perpendicular produce to V, so that let fall ST, which TV equal to ST ; may be and making the an and gles hsq, shq, equal to the angles VSP, SVP, about q as a centre, with an interval which shall be to ab as SP to VS, describe a circle cut ting the figure apb in p join sp, and draw : SH such that it may be to sh as SP is to sp, and may make the angle PSH equal to the angle psh, and the angle VSH equal to the Then with the foci S, H, and B angle pyq. principal axis AB, equal to the distance VH, I say, the thing is describe a conic section : done ; for if sv is drawn so that it shall be to

129 sp as sh is to sq, and shall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be similar, and therefore vh will be to pq as sh is to sq ; that is (because of the simi Wherefore triangles VSP, hsq), as VS is to SP ? or as ab to pq. vh and ab are equal. But, because of the similar triangles VSH, vsh, as vh to sh ; that is, the axis of the conic section now described is to lar VH SH the distance of is to now PSH P through the point ; as the axis ab to the distance of the foci foci its and therefore the figure because the triangle is described and because the pendicularly bisected by TR. line right VH three ; equal to its axis, and VS is per TR, the said figure touches the is line rght