4. Conics are drawn each touching two sides of a triangle at the angular points and intersecting in a point; prove that the intersections of the tangents at this common point with the sides cutting their respective conics lie on one straight line, and that the common tangents to the conics intersect the sides in the same three points.

5. A system of hyperbolas is described about a given triangle; prove that, if one of the asymptotes always pass through a fixed point, the other will always touch a fixed conic, to which the three sides of the triangle are tangents.

6. A parabola touches one side of a triangle in its middle point, and the other two sides produced; prove that the perpendiculars, drawn from the angular points of the triangle upon any tangent to the parabola, are in harmonical progression.

11. There is another system of Tangential Co-ordinates, which bears a close analogy to the ordinary Cartesian system. If x, y be the Cartesian co-ordinates of a point, referred to two rectangular axes, then the intercepts on these axes of the polar of the point, with respect to a circle whose centre is the origin, and radius k, will be

k2

х

k2

y

respectively. These intercepts

completely determine the position of the line, and their reciprocals may be taken as its co-ordinates, and denoted by the letters έ, n.

12.

In this system, every equation of the first degree represents a point.

be an equation of the first degree.

Draw the straight lines OX, OY at right angles to one another; on OX take the point A, such that OA = a, and on OY take the point B, such that OB=b. Draw AP, BP perpendicular to OX, OY respectively, meeting in P.

Then, the equation

a+bn = 1

shall represent the point P.

Through P draw any straight line, meeting. OX, OY in H, K, respectively. Then, if §, n be the co-ordinates of this line,

a relation which is satisfied by the co-ordinates of every line passing through the point P. This equation therefore represents the point P.

13. In this system, as in that described in the former part of the present chapter, an equation represents the curve, the

co-ordinates of whose tangents satisfy it, and an equation of the nth degree will therefore represent a curve of the nth class.

14. If the perpendicular OQ let fall from O on the straight line HK (fig. 23) be denoted by p, and the angle QOX by 4, we shall have

and a point will then be represented by the equation

a cos +b sin =p;

b

a

an equation which, if a2+b2 be put=c2, and =tana, becomes pc cos (4—a),

We thus obtain a method of representing curves by a relation between the perpendicular from a fixed point on the tangent and the inclination of that perpendicular to a fixed straight line. These may be called the tangential polar coordinates of the curve. This method will be found discussed in the Quarterly Journal of Pure and Applied Mathematics, Vol. I. p. 210.

EXAMPLES.

1. Prove that the distance between the points aέ+ by = 1, a'§+b'ŋ = 1, is {(a' — a)2 + (b' — b)°}§.

2. Prove that the cosine of the angle between the lines (§, 7),

3. Prove that the distance from the point (a+bn = 1) to the line (§,, n,) is (a§, + bn, − 1) (§,2 + n‚3) ̃§.

4. Prove that the equation & + n2 + 2P¢ + 2Qn+ R = 0, represents a conic, of which the focus is the origin.

What are the co-ordinates of its directrix? What is its eccentricity, and what its latus-rectum ?

5. Prove that the equation p = a + c cos represents a circle; and determine the radius of the circle.

6. Prove that the evolute of the ellipse a2¿2 + b2n2 = 1 is represented by the equation.

134

CHAPTER VIII.

ON THE INTERSECTION OF CONICS, ON PROJECTIONS, AND ON THE DETERMINATION OF A CONIC FROM FIVE GIVEN

GEOMETRICAL CONDITIONS.

1. WE shall here say a few words on the subject of the intersection of two conics, as an acquaintance with this branch of the subject will be useful in future investigations.

Since every conic is represented by an equation of the second degree, it follows that any two conics intersect in four points, which may be (1) all real, (2) two real and two imaginary, or (3) all imaginary.

2. Through these four points of intersection three pairs of straight lines can be drawn. If the four points be called P, Q, R, S, the pairs of straight lines will be PQ and RS, PR and QS, PS and QR. If PQ and RS intersect in L, PR and QS in M, PS and QR in N, the points L, M, N are called (see Art. 15, Chap. II.) the vertices of the quadrangle PQRS Also the three points L, M, N will form, with respect to every conic passing through the points P, Q, R, S, a conjugate triad; and therefore, each of them will have the same polar with respect to all such conics.

3. The equations of the pairs of lines PQ, RS, &c. (the sides and diagonals of the quadrangle) may be found as follows. Let the equations of the conics be

$ (a, B, y) = ua2 + vß2+ wy2+ Qu'ßy + 2v'ya+2w'aß = 0...(1), ↓ (a, B, y) = pa2+qB2 + ry2+ 2p'By +2q'ya + 2r'aẞ=0...(2); then every conic passing through their four points of intersection will be represented by an equation of the form

$(a, B, y) +ky (a, B, y) = 0.................................................(3).