Elliptic and hyperbolic trajectories

Lemma 15

If from the two foci S, H, of any ellipsis or hyberbola, we draw to any third point V the right lines SV, HV, where one HV is equal to the principal axis of the shape (the axis in which the foci are situated, the other SV is bisected in T by the perpendicular TR that falls on it. That perpendicular TR will touch the conic section somwhere. Vice versa, if it does touch it, HV will be equal to the principal axis of the shape.

Let the perpendicular TR cut the right line HV in R and then join SR. TS and TV are equal. Therefore the right lines SR and VR and the angles TRS, TRV, will also be equal.

Point R will be in the conic section, and the perpenpendicular TR will touch the same.

Proposition 18 PROBLEM 10

From a focus and the principal axes given, describe elliptic and hyperbolic trajectories, which shall pass through given points, and touch right lines given by position.

Let:

• S be the common focus of the shapes.
• AB is the length of the principal axis of any trajectory
• P is a point through which the trajectory should pass
• TR is a right line which it should touch.

The center P has an interval AB-SP. If the orbit is an ellipsis, or AB-SP, if the orbit is an hyperbola, describe the circle

About the centre P, with the

On the tangent TR let fall ST, and produce the same to V, so that TV may be V as a centre with the interval AB describe the and about to ST; equal In this manner, whether two points P, p, are given, or two circle FH. the perpendicular tangents TR, circles.

Let or a point P and a tangent TR, we are to describe two be their common intersection, and from the foci S, H, with tr, H the given axis describe the trajectory in the ellipsis, and cause -f- SP I say, the : PH PH SP thing is done. For (be in the hyperbola, is equal to the axis) the described trajectory will pass through the point P, and (by And by the same the preceding Lemma) will touch the right line TR. argument right lines it through the two points P, p, or touch the two will either pass TR.

Proposition 19 Problem 11

From a given focus, draw a parabolic passing through given points, and touch right lines given by position.

Let:

• S is the focus
• P a point
• TR a tangent of the trajectory to be drawn

With P as a centre, with the interval PS, draw the circle

FG. From the focus let fall ST perpendicular on the tangent, and may be equal to ST. produce the same to V, so as TV After the same manner another circle fg is to be de another point p is given or another point v is to be found, if another tangent tr is given; then draw the right line IF, which shall touch the two circles YG,fg, scribed, if if two points two the are or if two points V, v, P, p pass through given tangents TR, or touch the circle FG, and pass through the point V, if the tr, are given are given. On FI let fall the perpendicular point P and the tangent

SI, and bisect the same describe a parabola SK is : TR K in ; and with the axis SK and principal vertex K thing is done. For this parabola (because to FP) will pass through the point P and I say the equal to IK, and SP (by Cor. 3, Lem. XIV) because gle, it will touch the right line ST equal to is TR. 127 STR TV. and a light an