191. To find the angle subtended at the focus by the tangent drawn to a central conic from any point (xy).

Let the point of contact be (x'y'), the centre being the origin, then, if the radii from the focus F to the points (xy), (x'y'), be p, p', and make angles 0, 0, with the axis, it is evident that

but from the equation of the tangent we must have

xxyy
+ = 1.
α br

Substituting this value of yy', we get

or

pp′ co3 (0 − 6') = xx′ + cx + cx2 + c2 — — ¿xxx′ + b3,

a*

= e2xx′ + cx + cx' + a2 = (a + ex)(a + ex′) ;

or, since pa + ex', we have, (see O'Brien's Coordinate Geometry, p. 156),

a + ex
P

Since this value depends solely on the coordinates xy, and does not involve the coordinates of the point of contact, either tangent drawn from xy subtends the same angle at the focus. Hence, The angle subtended at the focus by any chord is bisected by the line joining the focus to its pole.

192. The line joining the focus to the pole of any chord passing through it is perpendicular to that chord.

This may be deduced as a particular case of the last Article, the angle subtended at the focus being in this case 180°; or directly as follows:-The equation of the perpendicular through xxyy any point x'y' to the polar of that point (

Art. 180,

a2

+

y = 1) is, as in

But if x'y' be anywhere on the directrix, we have x' = and

it will then be found that both the equation of the polar and that of the perpendicular are satisfied by the coordinates of the focus (x = c, y = 0).

When in any curve we use polar coordinates, the portion intercepted by the tangent on a perpendicular to the radius vector drawn through the pole is called the polar subtangent. Hence the theorem of this Article may be stated thus: The focus being the pole, the locus of the extremity of the polar subtangent is the directrix.

It will be proved (Chap. XII.) that the theorems of this and the last Article are true also for the parabola.

Ex. 1. The angle is constant which is subtended at the focus, by the portion intercepted on a variable tangent between two fixed tangents.

ByArt.191,it is half the angle subtended by the chord of contact of the fixed tangents.

Ex. 2. If any chord PP' cut the directrix in D, then FD is the external bisector of the angle PFP'. For FT is the internal bisector (Art. 191); but D is the pole of FT (since it is the intersection of PP', the polar of T, with the directrix, the polar of F); therefore, DF is perpendicular to FT, and is therefore the external bisector.

[The following theorems (communicated to me by the Rev. W. D. Sadleir) are

D

T

F

P'

founded on the analogy between the equations of the polar and the tangent.]

Ex. 3. If a point be taken anywhere on a fixed perpendicular to the axis, the perpendicular from it on its polar will pass through a fixed point on the axis. For the intercept made by the perpendicular will (as in Art. 180) be e2x', and will therefore be constant when z' is constant.

Ex. 4. Find the lengths of the perpendicular from the centre and from the foci on the polar of x'y'.

Ex. 5. Prove CM.PN' = b2. This is analogous to the theorem that the rectangle under the normal and the central perpendicular on

P

GT

M

N G

Ex. 7. Prove FG. F'G' CM.NN'. When P is

on the curve this theorem becomes FG. F'G' = b2.

193. To find the polar equation of the ellipse or hyperbola,

the focus F' being the pole.

The length of the focal radius vector (Art. 182) = a − ex′ ; but (being measured from the centre) = p cos + c.

The double ordinate at the focus is called the parameter; its half is found, by making 90° in the equation just given, to be

=

=a(1-e). The parameter is commonly denoted by the

The parameter is also called the Latus Rectum.

Ex. 1. The harmonic mean between the segments of a focal chord is constant, and equal to the semi-parameter.

For, if the radius vector FP, when produced backwards through the focus, meet

Ex. 2. The rectangle under the segments of a focal chord is proportional to the whole chord.

This is merely another way of stating the result of the last Example; but it may be proved directly by calculating the quantities FP. FP', and FP+ FP', which are easily seen to be respectively

Ex. 3. Any focal chord is a third proportional to the transverse axis and the parallel diameter.

For it will be remembered that the length of a semi-diameter making an angle with the transverse axis is (Art. 161)

2R2

Hence the length of the chord FP+ FP' found in the last Example

a

/Ex. 4. The sum of two focal chords drawn parallel to two conjugate diameters is

constant.

For the sum of the squares of two conjugate diameters is constant (Art. 173). Ex 5. The sum of the reciprocals of two focal chords at right angles to each other is constant.

194. The equation of the ellipse, referred to the vertex, is

Hence, in the ellipse, the square of the ordinate is less than the rectangle under the parameter and abscissa.

The equation of the hyperbola is found in like manner,

Hence, in the hyperbola, the square of the ordinate exceeds the rectangle under the parameter and abscissa.

We shall show, in the next chapter, that in the parabola these quantities are equal.

It was from this property that the names parabola, hyperbola, and ellipse, were first given (see Pappus, Math. Coll., Book VII.).

CONFOCAL CONICS.*

194 (a). Since the distance between the foci is 2c, where c2 = a* - b2, two concentric and coaxal conics will have the same foci when the difference of the squares of the axes is the same for both; and if we take the ellipse whose semi-axes are a and b, any conic will be confocal with it, whose equation is of the form

If we give the positive sign to X2, the confocal conic will be an ellipse; it will also be an ellipse when λ is negative as long as it is less than 6. When λ is between b' and a2, the curve will be a hyperbola, and when λ is greater than a", the curve is imaginary. If λ= b2, the equation reducing itself to y=0, the axis of x is itself the limit which separates confocal ellipses from hyperbolas. But the two foci belong to this limit in a special sense. In fact, through a given point can in general be drawn two conics confocal to a given one, since we have a quadratic to determine λ2, viz.

or λa − λ3 (a2 + b2 − x^ — y'") + a2b2 — b3x” — a3y” = 0.

When y = 0, this quadratic becomes (\" — b2) (λ2 — a* + x2) = 0, and one of its roots is 2, but if we have also x" a2 -l2,

*This section may be omitted on a first reading.

=

the second root is also λ2 = b2, and therefore the two foci are in a special sense points corresponding to that value of X".

If in the quadratic for λ we substitute λ=a, we get the positive result (a - b) x"; if we substitute λ=6' we get the negative result (62 — a2) y"; if we substitute negative infinity we get a positive result; hence, one of the roots lies between a2 and b2, and the other is less than 62; that is to say, one of the conics is a hyperbola and the other an ellipse, as is evident geometrically. In fact, through a given point P can clearly be described two conics having two given points F, F' for foci; viz. the ellipse, whose major axis is the sum of FP, F'P, and the hyperbola whose transverse axis is the difference of the same lines. Conversely, if a', a" be the semi-axes major of the ellipse and hyperbola, FP and F'P are a'+a” and a' - a".

194 (b). This theory can be made to furnish a kind of coordinate system which is sometimes employed; viz. any point Pis known when we know the axes of the two conics, confocal to a given one, which can be drawn through it; and in terms of these axes can be expressed the ordinary coordinates of P, and the lengths of all other lines geometrically connected with it. Perhaps the easiest way of getting such expressions is to investigate anew the problem of drawing through P a conic with given foci, taking for unknown quantity the transverse axis of the conic. Then since c2 is known, we write ac2 for b2, and have

In like manner, if b2 had been taken as the unknown quantity we should have had

The products of the roots of these equations are respectively cx and cy". Hence, we have at once expressions for the coordinates of the intersections of two confocal conics, viz. c"x"=a*a"", c'y"-b"b". The last value being negative, it-follows that one of the values of b2 is positive and the other