« ΠροηγούμενηΣυνέχεια »
(13.) The parabola referred to any diameter. A diameter and the tangent at its vertex being the axes, the equation is
y' = 4m,.,
or QV2=4 SP.PV.
If 4m, 4m, are the parameters corresponding to the ordinates Pp, Qq respectively, then
PM.Mp : QM.Mq:: m : mę.
If from the several points of any line given in position pairs of tangents be drawn to a parabola, the chords-joining the corresponding points of contact will pass through the same point. The chord of curvature through the focus=4 SP.
2 Spi The radius of curvature =
SA. (H. C. S. 71-84; Hust. Prop. 9.--15; Biot, 191-223;
G. G. A. Ch. xii.)
(14.) The Ellipse referred to its awes. Let S be the focus, and
PQ perpendicular to EQ the directrix; then
SP : PQ :: e : 1; e being a constant quantity, and < 1,
The equation referred to the axis, and a tangent at the vertex is
gø= (2ax– xo),
AS. Sa= BC = AC. SL.
y=ax +ß, may be determined from the equation
26*2 b?(B– aRa*) ya
aʻa? +62 which has only two roots, hence a straight line cannot cut an ellipse in more than two points.
aľa? + 72Y+
The equation to a tangent at the point (21,9,) is
CN.CT= AC; Cn.Ct + BC
The subtangent, NT=
2 1 If the ordinate NP be produced to meet the circumference of a circle described on Aa in R, the tangents at the points P and R will meet the axis produced in the same point. The equation to the normal PG is
(x — ,).
To draw a tangent to an ellipse from a given point (2392) without it: the point of contact (x,y) must be determined from the equations
62 a? (H. C. S. 89—111; Hust. Prop. 8–12; Biot, 133_60.) (15.) The ellipse referred to the focus. Let S, H, be the foci, then SP-a-ex,
SP + HP=2a= Aa.
AC – SC. cos PSN
1 1 2
1-(e cos 0) The angle SPT=HPt.
If SY, HZ be drawn perpendicular to the tangent, the points Y, Z are always in the circumference of a circle described on Aa.
SY.HZ=BC. (H. C. S. 112--22; Hust. Prop. 1–7.) (16.) The ellipse referred to any conjugate diameters.
If CD is conjugate to CP, conversely CP is conjugate to CD.
A tangent at P is parallel to CD.
The axes are the only conjugate diameters that are perpendicular to each other.
The equation to an ellipse referred to the centre and any conjugate diameters a,, b, is
If the extremity of the diameter is the origin, then
If any chords Qq, Rr, parallel to CD, CP, respectively, intersect each other in 0, then
Q0.09 : RO.Or :: CD : CP?.
az? +b = a + b?.
= 4CD. PF-4 AC. BC.
sin a ,b,=
tan aga =
(See Fig. p. 163.) AM .am=
= BC? Let Pp be the diameter drawn through any point P, whose co-ordinates are & q, Yı, and PQ, Qp the supplemental chords : the equation to PQ being
y-yı=a (x – 0,), the equation to Qp is
a 2a If the chords be drawn from the extremities of the axis major, their equations are
y=a (v + a);