Ex. 14. If from the vertex of an ellipse a radius vector be drawn to any point on the curve, find the locus of the point where a parallel radius through the centre meets the tangent at the point.

The tangent of the angle made with the axis by the radius vector to the vertex

; therefore, the equation of the parallel radius through the centre is

and the locus of the intersection of this line with the tangent

is, obviously,

x

a

=

1, the tangent at the other extremity of the axis.

The same investigation will apply, if the first radius vector be drawn through any point of the curve, by substituting a' and b' for a and b; the locus will then be the tangent at the diametrically opposite point.

Ex. 15. The length of the chord of an ellipse which touches a confocal ellipse, 2hb'2 the squares of whose semiaxes are a2 — h2, b2 — h2, is [Mr. Burnside]. ab The condition that the chord joining two points a, ẞ should touch the confocal conic, is

By the help of this Example several theorems concerning chords through a focus may be extended to chords touching confocal conics.

232. The methods of the preceding Articles do not apply to the hyperbola. For the hyperbola, however, we may substitute x' = a seco, y' = b tano,

This angle may be represented geometrically by drawing a tangent MQ from the foot of the ordinate M to the circle described on the transverse axis, then the angle QCM=4, since

M

We have also QM=a tano, but PM=b tano. Hence, if from the foot of any ordinate of a hyperbola we draw a tangent to the circle described on the transverse axis, this tangent is in a constant ratio to the ordinate.

Ex. If any point on the conjugate hyperbola be expressed similarly y" b seco', = a tan p', prove that the relation connecting the extremities of conjugate diameters is op'. [Mr. Turner.]

=

SIMILAR CONIC SECTIONS.

233. Any two figures are said to be similar and similarly placed, if radii vectores drawn to the first from a certain point are in a constant ratio to parallel radii drawn to the second from another point o. If it be possible to find any two such points O and o, we can find an infinity of others; for, take any point C, draw oc parallel to OC, and in the constant Q ratio op

ор

OP'

P

C

C

then from the similar triangles OCP, ocp, cp is parallel to CP and in the given ratio. In like manner, any other radius vector through c can be proved to be proportional to the parallel radius through C.

If two central conic sections be similar and similarly placed, all diameters of the one are proportional to the parallel diameters of the other, since the rectangles OP.OQ, op.oq, are proportional to the squares of the parallel diameters (Art. 149).

234. To find the condition that two conics, given by the general equations, should be similar and similarly placed.

Transforming to the centre of the first as origin, we find (Art. 152) that the square of any semi-diameter of the first is equal to a constant divided by a cos*0+2h cose sine+b sin2 0, and in like manner, that the square of a parallel semi-diameter of the second is equal to another constant divided by

a' cos2+2h' cose sin+b' sin" 0.

The ratio of the two cannot be independent of 0 unless

Hence, two conic sections will be similar, and similarly placed, if the coefficients of the highest powers of the variables are the same in both, or only differ by a constant multiplier.

235. It is evident that the directions of the axes of these conics must be the same, since the greatest and least diameters of one must be parallel to the greatest and least diameters of the other. If the diameter of one become infinite, so must also the parallel diameter of the other, that is to say, the asymptotes of similar and similarly placed hyperbolas are parallel. The same thing follows from the result of the last Article, since (Art. 154) the directions of the asymptotes are wholly determined by the highest terms of the equation.

be

Similar conics have the same eccentricity; for

=

m2a2- m2b2

m2a2

2

must

a2 - b2 a2 Similar and similarly placed conic sections

have hence sometimes been defined as those whose axes are parallel, and which have the same eccentricity.

If two hyperbolas have parallel asymptotes they are similar, for their axes must be parallel, since they bisect the angles between the asymptotes (Art. 155), and the eccentricity wholly depends on the angle between the asymptotes (Art. 167).

236. Since the eccentricity of every parabola is = 1, we should be led to infer that all parabolas are similar and similarly placed, the direction of whose axes is the same. In fact, the equation of one parabola, referred to its vertex, being y2 = px, or

p cose
9
sin20

it is plain that a parallel radius vector through the vertex of the other will be to this radius in the constant ratio p' : p.

Ex. 1. If on any radius vector to a conic section through a fixed point 0, OQ be taken in a constant ratio to OP, find the locus of Q. We have only to substitute mp for p in the polar equation, and the locus is found to be a conic similar to the given conic, and similarly placed.

The point / may be called the centre of similitude of the two conics; and it is obviously (see also Art. 115) the point where common tangents to the two conics intersect, since when the radii vectores OP, OP' to the first conic become equal, so must also 0Q, OQ' the radii vectores to the other.

Ex. 2. If a pair of radii be drawn through a centre of similitude of two similar conics, the chords joining their extremities will be either parallel, or will meet on the chord of intersection of the conics.

This is proved precisely as in Art. 116.

Ex. 3. Given three conics, similar and similarly placed, their six centres of similitude will lie three by three on right lines (see figure, page 108).

Ex. 4. If any line cut two similar and concentric conics, its parts intercepted between the conics will be equal.

Any chord of the outer conic which touches the interior will be bisected at the point of contact.

These are proved in the same manner as the theorems at page 181, which are but particular cases of them; for the asymptotes of any hyperbola may be considered as a conic section similar to it, since the highest terms in the equation of the asymptotes are the same as in the equation of the curve.

Ex. 5. If a tangent drawn at V, the vertex of the inner of two concentric and similar ellipses, meet the outer in the points T and T'', then any chord of the inner drawn through V is half the algebraic sum of the parallel chords of the outer through T and T'.

237. Two figures will be similar, although not similarly placed, if the proportional radii make a constant angle with each other, instead of being parallel; so that, if we could imagine one of the figures turned round through the given angle, they would be then both similar and similarly placed.

To find the condition that two conic sections, given by the general equations, should be similar, even though not similarly placed.

We have only to transform the first equation to axes making any angle with the given axes, and examine whether any value can be assigned to which will make the new a, h, b, proportional to a', h', b'. Suppose that they become ma', mh', mb'.

Now, the axes being supposed rectangular, we have seen (Art. 157) that the quantities a+b, ab-h2, are unaltered by transformation of co-ordinates; hence we have

a+b='m (a' + b'),

ab-h2 = m2 (a'b' — h2),

and the required condition is evidently

If the axes be oblique it is seen in like manner (Art. 158) that the condition for similarity is

(a+b− 2h cos w)* ̄ ̄ (a' + b' — 2h' cosw)2 *

It will be seen (Arts. 74, 154) that the condition found expresses that the angle between the (real or imaginary) asymptotes of the one curve is equal to that between those of the other.

THE CONTACT OF CONIC SECTIONS.

238. Two curves of the mth and nth degrees respectively, intersect in mn points.

th

For, if we eliminate either x or y between the equations, the resulting equation in the remaining variable, will in general be of the mn degree (Higher Algebra, p. 58; Todhunter's Theory of Equations, p. 169). If it should happen that the resulting equation should appear to fall below the mnth degree, in consequence of the coefficients of one or more of the highest powers vanishing, the curves would still be considered to intersect in mn points, one or more of these points being at infinity (see Art. 135). If account be thus taken of infinitely distant as well as of imaginary points, it may be asserted that the two curves always intersect in mn points. In particular two conics always intersect in four points. In the next Chapter some of the cases will be noticed where points of intersection of two conics are infinitely distant; at present we are about to consider the cases where two or more of them coincide.

Since four points may be connected by six lines, viz. 12, 34; 13, 24; 14, 23; two conics have three pairs of chords of intersection.

239. When two of the points of intersection coincide, the conics touch each other, and the line joining the coincident points is the common tangent. The conics will in this case meet in two real or imaginary points L, M distinct from the point of contact. This is called a contact of the first order. The contact is said to be of the second order when three of the points of intersection coincide, as for instance, if the point M move up until it coincide

T

T

T

M

with T. Curves which have contact of an order higher than the first are also said to osculate; and it appears that conics which osculate, must intersect in one other point. Contact of the third order is when two curves have four consecutive points common; and since two conics cannot have more than four points common, this is the highest order of contact they can have.