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be described touching the sides of a given triangle, but only one circumscribed about it. So, to have given that a conic is similar and similarly situated to a given one is equivalent to having two points given. To have given an asymptote is equivalent to having two points given, for an asymptote may be regarded as a tangent, the point of contact of which is given (at an infinite distance). To have given the direction of an asymptote is equivalent to having one point given, for this virtually determines the point in which the conic meets the line at infinity.

35. If it be given that three given points form a conjugate triad, this is equivalent to three conditions, as the equation of the conic, when these are taken as angular points of the triangle of reference, is of the form

ua+upa + wry = 0. Two more conditions will therefore completely determine the conic. If these conditions be that the conic shall pass through two given points, or touch two given straight lines, or pass through one given point and touch one given straight line, one conic only can be drawn to satisfy these conditions.

We may observe that, if the above conic pass through the point f, g, h) it also passes through the three points ff, g, h), (f, -9, h), (f. 9, h), and that, if it touch the line (i, m, n), it also touches the lines (-1, m, n), (1, - m, n), (1, m, -n).

EXAMPLES.

1. If XYZ be a triangle which moves in such a manner that its side Y Z always passes through a fixed point P, ZX through Q, XY through R, and if the locus of Y be a fixed conic passing through R and P, that of Z a fixed conic passing through P and Q, prove that the locus of X will be a fixed conic passing through Q, R, and through the other three points of intersection of the two given conics.

2. If two tangents be drawn to a conic so that the points in which they cut a given straight line form, with two fixed points on the straight line, a harmonic range, prove that the locus of their point of intersection will be a conic passing through the two given points.

3. A system of conics is described touching four given straight lines; prove that the locus of the pole of any fifth given straight line with respect to any conic of the system is a straight line.

If the fifth straight line be projected to infinity so that the points where it intersects two of the other given straight lines be projected into the circular points, what does this theorem become?

4. A system of conics is described about a given quadrangle; prove that the locus of the pole of any given straight line, with respect to any conic of the system, is a conic passing through the vertices of the quadrangle.

5. A system of conics is described touching the sides of a given triangle, and from a given point a pair of tangents is drawn to each conic of the system. · Prove that, if the locus of one of the points of contact be a straight line, that of the other will be a conic circumscribed about the given triangle.

6. The tangent at any point P of a conic, of which S and H are the foci, is cut by two conjugate diameters in T, t; prove that the triangles SPT, É Pt are similar to one another.

MISCELLANEOUS EXAMPLES.

1. Prove that the centre of the conic

+

+

1 1 1

0

aa * opcy coincides with the centre of gravity of the triangle of reference.

2. Prove that 0, 1, 1, 1 1=(x+y+z)(–yz)(y-%–)(2–2y).

1, 0, 2,y
1, 2, 0,
1, yo, x*, 0

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3. Prove that the square on the radius of the circle, described about the triangle of which the angular points are a, b, c, is

0, ab, ac
1 ba®, 0, bc
2 ca', cb, 0
0, 1, 1, 1
1, 0, ab, ac
1, ba, 0, bc
1, ca®, cb, o

Investigate a similar expression for the square on the radius of the sphere, described about the tetrahedron of which the angular points are a, b, c, d.

4. S is a focus of a conic, PQ a chord subtending a constant angle at S; SR, ST are drawn meeting the tangents at P and Q in R, T respectively, so that the angles PSR, QST are constant; prove that RT always touches a conic having S for a focus, and a directrix in common with the given conic.

5. Prove that, if the conic (la) + (mp)* + (ny) = 0 be a parabola, its focus and directrix are given by the equations

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Hence prove that, if a parabola touch three straight lines, its directrix always passes through a fixed point. State, in geometrical language, the position of this point, relatively to the three straight lines.

6., A system of parabolas is described so that a given triangle is self-conjugate with respect to each curve of the system; prove that the locus of the focus is a circle, that the directrix always passes through the centre of the circle described about the triangle, and that every parabola of the system touches the three straight lines which bisect each pair of sides of the triangle.

7. If P be any point on the circumference of a circle, 0 any fixed point, prove that the locus of the point, in which the tangent at Pintersects the line which bisects OP at right angles, is a straight line.

8. A rectangular hyperbola circumscribes a triangle; shew that the loci of the poles of its sides are three straight lines forming another triangle, whose angular points lie on the sides of the first, where they are met by perpendiculars from the opposite angular points.

9. If ABC, A'B'C' be two triangles, each of which is selfconjugate with regard to the same given conic, shew that another conic can be described about both.

+

10. If a, b, y, 8 be the distances of a point from four given straight lines, so connected that la + + ny + pd = 0, prove that, if a conic be described, touching these four straight lines, the locus of either of its foci will be the curve of the third degree represented by the equation

7

P р

0. βγ

т

n

+

+

a

11. Prove that the polar reciprocal of a rectangular hyperbola with respect to any point S, is a conic, the sum of the squares on the semi-axes of which is equal to the square on the distance of its centre from s.

12. Two given conics are so related that each of their common tangents subtends a right angle at a given point. Prove that, if any two points be taken, one on each conic, so that the line joining them also subtends a right angle at that point, the envelope of this line will be a conic, of which that point is a focus.

13. In Example 2, p. 116, prove that if any conic (A) be drawn touching the directrices of the four conics, the polar of the given point with respect to it will be a tangent to a conic, having the given point as focus and touching the sides of the triangle; and that the tangents from the given point to A are at right angles to each other.

14. If, through a fixed point 0, a straight line be drawn cutting the sides AB, AC of a triangle ABC in P, Q respectively, and BQ, CP be joined, prove that the locus of their point of intersection is a conic circumscribing the triangle ABC.

15. If Pa, por po be the semi-diameters of a conic, respectively parallel to the sides of the triangle of reference, prove that the area of the conic is

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16. PQ is the chord of a conic, having its pole on the chord AB or AB produced ; Q9 is drawn parallel to AB meeting the conic in q; shew that Pq bisects the chord AB.

17. Similar circular arcs are described on the sides of a triangle ABC, their convexities being towards the interior of the triangle; shew that the locus of the radical centre of the three circles is the rectangular hyperbola

sin (B-C) sin(C-A)
sin (C – A) sin (A B)

0.
ß

γ

+

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a

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