The elements of coordinate geometry, by S. L. Loney, M.A.

ANSWERS. xiii
XLVII. (Pages 397, 398.)
11, The locus can be shewn to be a straight line which is perpendicular to the given straight line; also the given straight line
touches one of the confocals and its pole with respect to that
confocal is its point of contact; this point of contact therefore
lies on the locus, which is therefore the normal.
14. As in Art. 366, use the Invariants of Art. 135.
XLVIII. (Pages 405-407.)
5. Two of the normals drawn from O coincide, since it is a centre of
curvature. The straight line 11x + my = 1 (Art. 412) is therefore
a tangent to the ellipse at some point 0 and hence, by Art. 412,
the equation to QR can be found in terms of 0.
XLIX. (Pages 414-416.)
1. (by-ax- c)2=4acx. 2. X2+y2-c (x+y)+ =-.
x2 y2
3. + 2=1 4. A parabola touching each of the two lines.
5. A central conic. 6. A parabola. 7. a2x2+b2y2=c4.
19. The line joining the foci is a particular case of the confocals and
the polar of 0 with respect to it is the major axis; the minor
axis is another particular case, so that two of the polars are lines
through C at right angles; also the tangents at O to the confocals through it are two of the polars, and these are at right
angles. Thus both C and O are on the directrix.
21. The crease is clearly the line bisecting at right angles the line
joining the initial position of C to the position which C occupies
when the paper is folded.
23. cos a cos
23. I =1 ~ e cos a cos 0.
7'