EXAMPLES.

1. If CP be a semi-diameter of an ellipse, and AQQ' be drawn parallel to CP, meeting the curve and CB in Q, Q', then 2 CP" AQ.AQ.

=

2. What parallelogram circumscribing an ellipse has the least area?

3. If straight lines drawn through any point of an ellipse to the extremities of a diameter meet the conjugate CD in M, N, then CM.CN=CD2.

4. If two tangents to an ellipse and the chord of contact include a constant area, the area included between the chord of contact and the ellipse is constant.

5. Prove also that the chord of contact always touches a concentric similar ellipse, and that the intersection of the tangents lies on another concentric similar ellipse.

6. If CP, CD be conjugate and AD, A'P meet in O, then BDOP is a parallelogram. When is its area greatest?

7. From the ends, P, D, of conjugate diameters in an ellipse draw lines parallel to any tangent line, and from the centre C draw any line cutting these lines and the tangent in points p, d, t; then will Cp+ Cd2 = Ct".

8. The least triangle circumscribing a given ellipse has its sides bisected at the points of contact.

9. If an ellipse be inscribed in a given parallelogram its area will be greatest when the sides are bisected at the points of contact.

10. A polygon of a given number of sides is described about an ellipse and has its sides bisected at the points of contact. Prove that its area is constant.

11. Prove also that if the adjacent points of contact be joined the area of polygon thus formed will be constant.

12. If a triangle be inscribed in an ellipse, the straight lines drawn through the angular points parallel to the diameters bisecting the opposite side meet in a point.

13. The greatest triangle which can be inscribed in an ellipse has one of its sides bisected by a diameter of the ellipse and the others cut in points of bisection by the conjugate diameter.

14. The tangent and ordinate, at any point of an ellipse, meet the axis in T, N. Prove that

AN.A'N: AT.A'T-CN: CT.

15. Circles correspond to similar and similarly situated ellipses.

16. Parallel straight lines which pass through the extremities of conjugate diameters meet the ellipse again at the extremities of conjugate diameters.

17. Two ellipses of equal eccentricity and whose major axes are equal can only have two points in common. Prove this, and show that if three such ellipses intersect, two and two, in the points P, P'; Q', Q'; R, R', the lines PP', QQ', RR' meet in a point.

18. The locus of the middle points of all focal chords in an ellipse is a similar ellipse.

19. The locus of the middle points of all chords of an ellipse which pass through a fixed point is a similar ellipse.

20. The greatest triangle that can be inscribed in an ellipse has its sides parallel to the tangents at the opposite vertices.

21. P, Q, R are any three points on an ellipse; the diameter ACA' bisects PQ and meets RP, RQ in N, T. Prove that CN.CT-CA".

22. From an external point two tangents are drawn to an ellipse; show that an ellipse, similar and similarly situated, will pass through the external point, the points of contact, and the centre of the given ellipse..

23. A and B are two similar, similarly situated, and concentric ellipses; C is a third ellipse similar to A and B, its centre being on the circumference of B, and its axes parallel to those of A or B. Show that the common chord of A and C is parallel to the tangent to B at the centre of C.

24. Any chord of a conic which touches a similar, similarly situated and concentric conic is bisected at the point of contact.

25. The two portions of any straight line intercepted between two similar, similarly situated, and concentric conics, are equal.

26. A tangent to the interior of two similar, similarly situated, and concentric ellipses, cuts off a constant area from the exterior.

27. Through a given point draw a straight line cutting off a minimum area from a given ellipse.

28. If a chord of an ellipse pass through a fixed point, pairs of tangents at its extremities will intersect on a fixed straight line.

29. A chord of an ellipse, drawn through any point, is cut harmonically by the point, the curve, and the polar of the point.

30. If a tangent drawn at V the vertex of the inner of two concentric, similar, and similarly situated ellipses, meet the outer in the points T, T", then any chord of the inner, drawn through V, is half the sum, or half the difference of the parallel chords of the outer through T, T".

.

CHAPTER X.

CURVATURE.

Let a circle be described touching a conic at P and cutting it in an adjacent point Q. Then, when Q moves up and ultimately coincides with P, the circle becomes the Circle of Curvature at the point P.

The chord of this circle drawn in any direction from P, is said to be the Chord of Curvature at P in that direction.

The radius, diameter, and centre of the circle of curvature are called respectively the Radius, Diameter and Centre of Curvature.

PROP. I. The focal chord of curvature at any point of a conic is equal to the focal chord of the conic parallel to the tangent at that point.

Let PSP' be any focal chord of the conic; and RR' the focal chord parallel to the tangent PT.

Let a circle be described touching the conic at P and cutting it in Q. Also let QH, a chord of this circle, parallel to PP', meet PT in T.

Then, if TQ meet the conic again in Q',

But

TP: TQ.TQRR': PP'.

[Cor. 3, p. 85.

TP-TQ.TH.

[Euc. III., 36.

Hence

Let

TH:TỰ = RR': PP'.

move up to P. Then TQ' becomes equal to PP',

and the circle becomes the circle of curvature at P.

Hence the focal chord of curvature PU, to which TH becomes equal, is equal to RR'.

COR. 1. Hence

PU.SE=2PG3,

[Ex. 21, p. 21.

where PG is the normal at P, SE the semi-latus rectum, and PU the focal chord of curvature as in the proposition.

COR. 2. In a central conic

PU.CA-2CD".

[Ex. 26, p. 92.

PROP. II. To determine the length of the chord of curvature of a parabola drawn in any direction.

Let PSU be the focal chord of curvature at P, and PV a chord of curvature drawn in any other direction. [fig., Prop. III.

Join UV, and draw SY parallel to VP to meet the tangent at P in Y. Then the triangles UPV, PSY are similar, since the alternate angles UPV, PSY are equal and SPY is equal to PVU in the alternate segment.

But PU is equal to the focal chord of the parabola parallel to the tangent at P (Prop. 1.) that is, to 4 SP. [Prop. VIII., p. 29.

Hence

or

PV: 4SP=SP: SY,

PV.SY=4SP.