Ex. 7. Straight lines move so that the triangular area which they cut off from two given straight lines which meet one another is constant: to find the locus of their ultimate intersections.

Let OAA', OBB' be the fixed lines, AB, A'B' two of the moving lines with the condition that

OA.OB=OA'. OB'.

If a, ẞ be unit vectors along OA, OB,

OA=ta, OB=uß; OA' t'a, OB = u'ß,

the point of intersection of AB, AB′ gives

=c because the triangle has a constant area;

2. If the tangent to a parabola cut the directrix in R, SR is perpendicular to SP.

3. A circle has its centre at the vertex A of a parabola whose focus is S, and the diameter of the circle is 3AS. Prove that the common chord bisects AS.

4. The tangent at any point of a parabola meets the directrix and latus rectum in two points equally distant from the focus.

5. The circle described on SP as diameter is touched by the tangent at the vertex.

6. Parabolas have their axes parallel and all pass through two given points. Prove that their foci lie in a conic section.

7. Two parabolas have a common directrix. Prove that their common chord bisects at right angles the line joining their foci.

8.

The portion of any tangent to the parabola between tangents which meet in the directrix subtends a right angle at the focus.

9. If from the point of contact of a tangent to a parabola a chord be drawn, and another line be drawn parallel to the axis meeting the chord, tangent and curve; this line will be divided by them in the same ratio as it divides the chord.

10. The middle points of focal chords describe a parabola whose latus rectum is half that of the given parabola.

11. PSQ is a focal chord of a parabola: PA, QA meet the directrix in y, z. Prove that P, Qy are parallel to the axis.

12. The tangent at D to the conjugate hyperbola is parallel to CP.

13. The portion of the tangent to a hyperbola which is intercepted by the asymptotes is bisected at the point of contact.

14. The locus of a point which divides in a given ratio lines which cut off equal areas from the space enclosed by two given straight lines is a hyperbola of which these lines are the asymptotes.

15. The tangent to a hyperbola at P meets an asymptote in T, and TQ is drawn to the curve parallel to the other asymptote. PQ produced both ways meets the asymptotes in R, R': RR' is trisected in P, Q.

16. From any point R of an asymptote, RN, RM are drawn parallel to conjugate diameters intersecting the hyperbola and its conjugate in P and D. Prove that CP and CD are conjugate.

17. The intercepts on any straight line between the hyperbola and its asymptotes are equal.

18. . If QQ' meet the asymptotes in R, r,

19.

RQ. Qr = PO.

If the tangent at any point meet the asymptotes in X and Y, the area of the triangle XCY is constant.

CHAPTER VIII.

CENTRAL SURFACES OF THE SECOND ORDER, PARTICULARLY THE ELLIPSOID AND CONE.

56. The Ellipsoid. In discussing central surfaces of the second order, we shall speak as if our results were limited to the ellipsoid. That such limitation is not, in most cases, necessarily imposed on us, will be apparent to any one who has a slender acquaintance with ordinary Analytical Geometry. We adopt it in order that our language may have more precision, and that, in some instances, our analysis may have greater simplicity. If the centre be made the origin it is clear that the scalar equation can contain no such term as ASap, for the definition of a central surface requires that the equation shall be satisfied both by +p and by - p.

If we turn to the equation of the ellipse (Art. 43), we shall see at once that the equation of the ellipsoid must have the form ap2+bS3ap + 2cSapSẞp + ... = 1.

Now if, as in the Article referred to, we put

It will be seen that, as in Arts. 32, 33, one form of the equation of the straight line was found to coincide exactly with the equation of a plane, so a form of the equation of the ellipse coincides exactly with the equation of the ellipsoid.

It is evident that the three properties of op given in Art. 44 are true of op in its present form.

57. To find the equation of the tangent plane.

Let a secant plane pass through the point whose vector is p; and let p' be the vector to any point of section.

then

and

Put p' = p + ß, where ẞ is a vector along the secant plane;

S'p'op' = S (p + B) & (p + B).

Hence, observing that (44)

• (p + ß) = $p + &ß,

we have

Σρφβ = βφρ,

Sp'op' = Spop + 2Sßop + Sß&ß ;

i.e. 2.Sẞop + SB&B = 0.

Now (45), as the secant plane approaches the tangent plane, the sum of these two expressions approaches in value to the first alone: that is, for the tangent plane, Sẞøp = 0, where ẞ is a vector along that plane.

and

If be the vector to a point in the tangent plane,

is the equation of the tangent plane.

COR. op is a vector perpendicular to the tangent plane at the extremity of the vector p.

58. If Or be perpendicular from the centre O on the tangent plane; then, since op is a vector perpendicular to that plane, OY= xop and Sx (pp)2 = 1, giving

Sir W. Hamilton terms op the vector of proximity. [In fact vector OY= (pp)']

T. Q.

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