one another in M; also let OA be a semi-diameter parallel to PQ, and OG one parallel to P'Q': then

PM.MQ P'M. MQ':: OA2 : OG2.

Let OA', OG' be semi-diameters conjugate to OA and OG respectively; the chords PQ, P'Q' being consequently bisected by those semi-diameters, and the former in the point R. Through M draw the semi-diameter OK, and draw KT parallel to AO.

Let OA'a, OA=b, OR = x, RP = y;
OT, TK = y.

also

By the similarity of the triangles OKT, OMR,

and, for the point м, each of these ratios is constant; let either of them be represented by m. Then,

Now, in the equation (2) multiplying b2, in the first term,

and y'2 by m2, and the second term of the first member by

(the equivalent of m2) that equation becomes

And subtracting this last equation from (1) we have

b2-b2 m2 = y2-m2y'2.

(3)

x2

x/2

But the second member of this equation is RP2 - MR2; which, since PQ is bisected in R is (Euc. 5. II.) equivalent to

PM.MQ.

Thus

PM.MQ=62 (1 — m2).

In like manner it may be shown, representing OG by b', and drawing an ordinate KT' from K to the diameter OG', that P'M. MQ'b'2 (1—m2);

Consequently

PM.MQ P'M. MQ :: b2: b'2,

or the rectangles of the segments of the chords are to one another as the squares of the semi-diameters which are parallel to the chords.

M

a. If the point м were on the exterior of the ellipse, on subtracting (1) from (3) we should have MR2-RP2 (=PM. MQ) = b2 (m2-1); and in like manner,

P'M. MQ'b'2 (m2-1):

Consequently the rectangles of the produced chords, and the parts beyond the ellipse, are to one another as the o squares of the semi-diameters which are parallel to such chords.

K

Again, if the point м were on the exterior of the ellipse, and a line MPQ were drawn through, while another line MP" were to touch the ellipse; we should, as before, b and b' representing the semi-diameters parallel to MQ and MP", get PM. MQ-b2 (m2-1):

while, after multiplying the equation corresponding to (2) above by m2 or its equivalent, we should have

Therefore,

MP2 b'2 (m2-1).

=

PM. MQ MP/2 :: b2 : b'2.

And it is easy to perceive that, if two lines MP", MQ" were drawn from one point м to touch the ellipse, the squares of those tangents would be, to one another, as the squares of the semi-diameters of the ellipse, which are parallel to them.

The demonstration of the proposition, in all its cases, would be the same for an hyperbola, on employing, instead of (1) and (2) the corresponding equations for that curve.

b. With respect to a parabola: since the diameter oa', which bisects PQ in R, is parallel to MK, the diameter drawn through M, we have

MR KT.

But A'R and A'T being represented by x and x', also RP by y, and TK or MR by y'; and again, p being the parameter of the diameter OA', we have (Art. 72.)

therefore,

But x

p (x—x')=y2—y 2.

x' is constant for the point м; let this constant be represented by m; then

In like manner, p' being the parameter of the diameter O'G' to which P'q'is a double ordinate,

p'm=P'M. MQ'.

Thus the rectangles of the segments of the chords are to one another as the parameters of the diameters of which the chords are double ordinates; or (Art. 108.) as the distances of the focus from the vertices of those diameters.

By similar processes it may be proved that, in a parabola, the rectangles of the segments of lines drawn from a point on the exterior, or the squares of the tangents drawn from a point on the exterior, are to one another as the squares of the parameters of diameters parallel to the lines.

PROPOSITION XXIX.

126. If chords are drawn in an ellipse, hyperbola or parabola, to intersect one another on any diameter, and tangents are drawn from the extremities of each to intersect one another, the points of intersection for every pair of tangents will be in one straight line parallel to a tangent at the vertex of that diameter.

Let PQ in both figures be a chord intersecting the diameter AB, of which it is a double ordinate, in R, and let the tangents at P and Q meet (Art. 97.) in T. Let P'Q' be any other chord passing through R, and let P'T', Q'T' be tangents at P' and Q, meeting in some point r'; then T' will be in a straight line, drawn through T parallel to A'B', which is conjugate to A B, or parallel to a tangent to the curve at A. Let OA (a),

OA' (b) be considered as the co-ordinate axes; let x and y (=OR' and R'P') represent the co-ordinates of P', and let x, y be the co-ordinates of any point in P'T'.

Then, for an ellipse, the equation for a tangent as P'T', is [(a') Art. 90.],

a2yy+b2xx'=a2b2,

and for the tangent Q'T',

— a2y,y′′ +b2x,x" — a2 b2,

x" and —y" (=OR" and R′′Q ́) being the co-ordinates of q', while x, and y, are those of any point in Q'T'.

But, at T' the point of intersection, x, y, are respectively equal to x and y; therefore, subtracting the first equation from the second,

Now, from the similarity of the triangles RR'P', RR”Q', we have RR RR" :: R'P': R′′Q';

whence, by conversion, RR': R′R" :: R′P′ : R′P'+R'Q'′,

But the second member of this equation is equivalent to x'-x' ;; therefore putting OR-OR', or OR-x', for RR', we

y"+y''

Thus x, or OT, is constant for all chords passing through R, and it follows that the intersection T' of the pairs of tangents to the ellipse, at the extremities of such chords, is in a line TT' passing through T parallel to A'B' or AY.

If AB were the transverse axis, and R the focus of the ellipse, it is evident (Art. 88.) that all the points T, T', &c. would be in the directrix.

In like manner may the proposition be demonstrated for an hyperbola, in which the equation of a tangent is [(') Art. 90.]

b2xx' — a2yy' = a2b2.

For a parabola, let MAN be part of the curve; then the equation for a tangent as P'T' being [(c") Art. 90.]

2yy'=px+px',

in which, with reference to the diameter A B, x' and y' are the co-ordinates of P', the point of contact, x and y the coordinates of any point in P'T', and p the parameter of AB; also the equation for Q'T' being

the T

by subtraction we have, for the point T at which x, x and y,=y,

whence

2 (y'+y') y=p (x'—x'');

2yx'-x"

=

Р y' +y"

T

(=, by similarity of triangles,

Then, for R'R putting AR' - AR, or x'-AR, we have

putting for 2yy' its equivalent px+px' above, the equation becomes

XAR.

Thus for the point T' of intersection, x is constant when the chords pass through R; and therefore all the pairs of tangents intersect one another in a line passing through T parallel to AY, or to any of the ordinates of A B.

If AB were the axis and R the focus of the parabola, it is evident (Art. 88.) that all the pairs of tangents would meet on the directrix.

127. COR. In a parabola, the two tangents drawn from the extremities of every chord passing through the focus will meet on the directrix at right angles to one another.

It has been already proved that tangents so drawn intersect one another on the directrix. Let TT' be the directrix, which will then be at right angles to AB, the axis; also, let P'T', Q'T' be two such tangents, and A'B' a diameter drawn through T'; then, since tangents drawn from the extremities of a double ordinate to that diameter meet on that diameter produced, P'Q' is a double ordinate to A'B', and is consequently bisected in R,. Now A' being the vertex of that diameter, and R the focus of the parabola, T'A', A'R,, and A'R are each equal (Art. 72.) to one quarter of the parameter of A'B', while R, P' and R, Q' are each equal to half that para