28. The tangent from P a point in the asymptote touches the curve in 0 : HT is parallel to the same asymptote; prove that HP bisects the angle THO: and if PQ intersects the other asymptote in Q, PHQ-half the angle PCQ.

29. The focal distances of two points P, P' intersect in 0, prove that the tangents QP, QP subtend equal angles at O.

30. If SP, HQ are parallel, find the locus of the intersection of the tangents at P and Q.

31. PT, QT tangents from Ta point in the auxiliary circle whether to the same branch or not have one of the focal distances SP, SQ parallel to one of the two HQ, HP

32. The tangent at P is perpendicular to two parallel tangents at Q, Q', prove that SQ, HQ' subtend equal angles at P.

33. If GL be the perpendicular from G on SP, prove GL : PN a constant ratio.

34. From the vertex A draw AQ perpendicular to the tangent at P, and let QA produced meet PS produced in O, the locus of O is a circle.

35. Let GYproduced meet ZH produced in Z', then HZ'=HZ.

36. If the normals at P, Q extremities of a focal chord intersect in O, and OL parallel to the transverse axis cut SP in L, L is the middle point of PQ.

37. If GR be the perpendicular on SP from G, PR equals semi-latus rectum.

38. If the normal at P meets the axes in G, g, the triangles SPg, GSg are similar, and Gg: Sg and therefore Pg: Sg are constant ratios.

CHAPTER VIII.

On the Parabola.

1. WE have now to consider the case in which the cutting plane is parallel to one of the generating lines of the cone. Let the cutting plane intersect the cone in

P

P

R

the curve PAP, and intersect the plane of the paper which

contains the axis in the line AN parallel to the generating line VR in the plane of the paper, then PAp is called a Parabola.

It is manifest that the parabola is divided into equal parts by the line AN which is called the axis, every line in the plane of the section at right angles to AN will meet the curve in two points on opposite sides of AN and at equal distances from it,

The axis intersects the curve in one point only, in other words the curve has only one vertex.

2. By turning the cutting plane about a line through A at right angles to the plane of the paper through the smallest possible angle, the parabola is changed either into an ellipse or hyperbola whose centre is at a great distance from C: the parabola may therefore be considered as the form to which the semi-ellipse cut off by the axis minor on the side of A and one branch of the hyperbola approach more and more nearly when, the vertex remaining fixed, the centre is removed to a greater and greater distance. Hence it appears that lines drawn from all points in the semiellipse and semi-hyperbola to their centres become more and more nearly parallel as the cutting plane moves towards the position in which it cuts a parabola from the cone.

And we may anticipate that the line joining the middle points of parallel chords of a parabola will be parallel to its axis; and generally all properties of the ellipse and hyperbola that relate to lines that remain finite in the parabola are equally true in that curve.

It will have been observed that there is a similarity in the properties of the ellipse and hyperbola; the parabola is the curve in which they meet, or in which each undergoes the transition into the other.

3. NP2-4AS. AN.

Let RPR be a circular section of the cone made by a plane at right angles to the axis through N, any point of

the axis of the parabola and cutting the plane of the curve in PNP at right angles to the axis.

Then NP2= RN. NR', and the ratio NR: AN is constant for all positions of N.

Hence the ratio NP2: AN. NR' is constant: and NR' is also invariable.

The ratio RN: AN is invariable for all positions of N; the ratio NR: AN may be made to have all values from a ratio indefinitely great to one indefinitely small by moving N along the axis from A. Hence we may find a point S in the axis such that for that point RN. NR' = 4AN2, or if LSL be the double ordinate through S, SL 4AS2 or SL=2AS.

=

Hence generally

NP SL: AN. NR': AS. NR AN AS,

NP: 4AS: AN: AS,

.. NP2=44S. AN

4. QR. QR=4AS. PR.

Let PR from any point P of the curve meet the double ordinate QMQ'in R, then QR. Q'R = 4AS. PR. Draw the

double ordinate PNP', then PM is a rectangle whose opposite sides are equal: and since QQ' is bisected in M,

QR. Q'R+RM2 = QM2,

or QR. QR + PN2 = QM2;

.. QR. Q'R + 4AS. AN = 4ASAM;

.. QR. Q'R 4AS. MN

=

= 4AS. PR.