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16. a;2 + 7/2-22a;-4?/ + 25 = 0. 17. x'^ + i/-5x-y + ^=0.

18. 3a;2 + 3?/2-29a;-19?/ + 56 = 0.

19. & (a;2 + 7/2) _ (a2 + &2) ^ + (a _ 6) (a2 + 1-) = 0.
21. a;2 + ?/2_3a;_42/=0.



VI COORDINATE GEOMETKY.

23. cc'^ + y^ - hx-ky = 0. 24. x^ + if-2ijja^-b^=b'^.

25. a;2 + 2/2_iOa;-10?/ + 25 = 0. 26. a;2 + 2/2-2aa;-2a?/ + a2=0.

27. ^2 + 2/2 + 2(5^^12) {a; + 2/) + 37±10Vl2 = 0.

28. x^ + y^-ex + 4y + 9 = 0. 29. 6 (a;2 + 2/2)=a; (ft^ + c^).

30. a;2 + i/2+6V22/-6a; + 9 = 0.

31. x^ + y^-3x + 2 = 0; 2x^ + 2y^-5x- ^Sy + S = 0;

2x^ + 2y^-7x-jSy + Q = 0.

33. {a;+21)2 + (i/ + 13)2=652. 34. 8a;2 + 8y2 - 25a; -3^/ + 18=0.

36. a;2 + i/2=a2 + &2; x'^ + y^-2{a + b)x + 2{a-b)y + a'^ + h^ = 0.

XVin. (Pages 134, 135.)

1. 5a; -12^ = 152. 2. 24a; + 10?/ + 151 = 0.

3. a; + 2i/=±2V5. 4. x + 2y +g + 2f= ±J5 ^g^+f-c,

5. [-^^ ;j2)' ^- c = a;(0, &). 7. Yes.

8. fc = 40or-10. 9. a cos2 a + b sin2 a i s/a^ + &2 sin2 a.

10. Aa + Bb + C=:i=c >Ja^+B\

11. (1) y = mx±ajl + m^; (2) m?/ + a; = ± a ^/f+m^ ;
(3) ax±y Jb^-a?=ab', (4) a; + i/ = aV2.

12. 2^?-2-^2-p^2. 13. a;2 + ,/±V2aa; = 0; a;2 + 2/2±V2«?/ = 0.

14. c = 6-a7?i; c = b-am^ sj{l + m'^){a'^+b'^).

15. aj^ + ?/2-6a;-8?/+ff^ = 0.

16. a:2 + ?/2_2ca;-2c?/ + c2=0, where 2c=ia + b^ ^aFVb\

17. 5a;2 + 5?/2- 10a; + 30^ + 49 = 0. 18. a;2 + 2/2- 2ca;-2c?/ + c2=0.
19. {x-if-V{y-hY=r''. 20. x'' + y^-2ax-2^y = 0.

XIX. (Pages 144, 145.)

1. x + 2y = 7. 2. 8a;-2y = ll. 3. x = 0.

4. 23a; + 5?/ = 57. 5. by-ax = a\ 6. (5,10).
7. (I. -tU 8. (1,-2). 9. (i, -i).

10. (-2a, -26). 11. (6, - V-).

12. 3?/-2a;=13; (-W-,-W)- 13. (2,-1). 14. a;'2 + ^'23,2a2.

18. iV46. 19. 9. 20. V2a2 + 2a& + 62. 21. (¥, 2) ; |.
23. (1) 28a;2 + 33a;?/ -28?/2- 715a; -195^ + 4225 = 0;

(2) 123x2 - Uxy + 3?/2 _ C64a; + 226?/ + 763 = 0.



ANSWERS. Vll

XX. (Pages 147, 148.)

2. r- - 2ra cosec a . cos (^ - a) + a^ cot^ a = 0, r = 2a sin 6.
6. r2-r[acos(^-a) + &cos(^-jS)] + a6cos(a-j8) = 0.
8. &V + 2ac=:l.

XXI. (Page 149.)

1. 120O; (^-^^4^); ^-fV/^^^T^..

2. 30°; (8-6^3, 12-4^3); J^l -24.^%.
f g-fcoSQ} f-g cos o} \ ^ Jp + g^-2fgG0S(a
\ siu-^ w sm-^ w / sm w

4. x^- + ^2xy + y^ - x{4: + 3^2)-2ij{3 + ^2) + S{J2-l)=0.

5. a;2 + a;?/ + 2/2 + 11a; + 13?/ + 18 = 0.

8. (a; - x') {x - x") + {y - y') (y - y") + cos w [{x - x') {y - y")

+ {x-x"){y-y')] = 0.

XXII. (Pages 156—159.)
4. A circle. 5. -A. circle. 6. -A. circle.

ft SlU 03

9. a;^ + 7/2 - 2a;?/ cos w= ^r — , the given radii being the axes.

11. A circle. 12. A circle.

16, (1) A circle ; (2) A circle ; (3) The polar of 0.

17. The curve ?' = a + acos^, the fixed point being the origin and
the centre of the circle on the initial line.

24. The same circle in each case.

33. 2ah^sJa^TV\ 35. a Vtt ; ^ = 4a; 63a;+16?/ + 100a = 0.

36. (i) x = Q, 3a; + 4i/ = 10, 2/ = 4, and 3?/ = 4a;.
(ii) 2/ = wia;+ c ^/l + ??^'^, where

±(& + c) ±(6-c)



XXIII. (Pages 164, 165.)

3. 3a;2 + 3?/2-8.v + 29?/ = 0. 4. lDX-lly = lU.

5. x + mj = 2. 6. 6x-7y + 12 = 0. 7. (-1,-1).

8. (If.ii^). 11. (A + l)(a;2 + 2/2) + 2\(a; + 2?/) = 4 + 6\.

13. {y-xf=0.



Vlll COORDINATE GEOMETRY.

XXIV. (Pages 172, 173.)

8. x^-y^ + 2mxy = c. 12. k{x^ + y'^) + {a-c)y-ck=0.

13. x^- + 7f-cx-by + a^=0. 14. x^ + y^-lQx-18y -4: = 0.

XXV. (Pages 178, 179.)

1. (7a; + 6?/)2- 570a; + 750?/ + 2100 = 0.

2. {ax- lyf - 2a^x - 2hhj + a^ + a%^ + 64=0.

3. (-1,2); y = 2', 4; (0,2). 4. (4, |) ; a; = 4; 2; (4,4).

5. («'|); ^ = «; 2a; (a, 0). 6. (1,2); 2/ = 2; 4; (0,2).

8. (i)i; (ii) 4. 9. (2,6). 11. y^-2x',y-12 = m{x-2i).

XXVI. (Pages 185—187.)

1. 4?/ = 3.'c + 12; 4a; + 3i/ = 34. 2. 4?/-a; = 24; 4a; + 2/=108.

3. 2/~^ = ^' y+x = ^', a; + 2/ + 3 = 0; a;-?/ = 9.

4. y = x; x + y = 4a; y + x = ; x-y = 4:a.

5. 4i/ = a; + 28; (28,14). 6. (|, ^) .

7. 2/ + 2^ + l = 0; (i, -2); 2z/ = a; + 8; (8,8).

8. (3a, 2^3a); ^|, -^aV 9. 4?/ == 9a; + 4 ; 4?/ = a; + 36.

13. (^^^«» aV275 + 2^; (3a, 2^3a).

14. h^y + a\v + aibi = 0. 15. a;=0.

XXVII. (Pages 197, 198.)

4. ix + Sy+l = 0. 5. 56t/ = 25.

XXVIII. (Pages 203—205.)

25. Take the general equation to the circle and introduce the
condition that the point (at'^, 2at) lies on it ; the sum of the
roots of the resulting equation in t is then found to be zero.

28. It can be shewn that the normals at the points "^i" and "ig"
meet on the parabola when tjt^=2 ; then use the previous
example.

XXIX. (Pages 209—211.)

1. y = bx. 2. cx = a. 3. y = ad.

4. y = [x-a) tan 2a. 5.2/^- ^^^ = 2ax.

6. x^ = iuL^i{x-af+y^]. 19. y^ = 2a{x-a).



^. 7. 7x^ + 2xy + 7y^ + 10x-10ij + 7 = 0. 8. Without.



ANSWERS. IX

20. y^-ky = 2a{x-h). 21. tf{tf-2ax + 4.a^-) + Sa'i^=0.

22. (8a2 + i/2_2aa;)2tan2a = 16a2(4aa;-?/).

23. y^ + 4:ay^{a-x)-lQa^x + an'^=0.

24. The parabola t/^ = 2a{x + 2a).

XXX. (Pages 214—216.)
1. y^ = a{x-a), 2. y^ = ^ax. 3. 27ay2 = (2.r ~ rt)(a;-5a)2.

4. A parabola. 5. A straight line.

6. 27at/^ -4 (a; -2aP=: constant. •

7. A straight line, itself a normal.

XXXII. (Pages 234, 235.)

1. (a) 3a;2 + 5?/=32; (/3) 3a;2 + 7?/ = 115.

2. 20a;2 + 36i/2 = 45. 3. x'^ + 2y'^=100. 4. 8.^2 + 9^/2= 1152.

5. (l)y; W6; (^iV6,0); (2)1; W^; (0,±tVV5);

(3) -V-; I; (0, 5) and (0, 1).

./3

9. x + 4V3t/ = 24V3; 11a; -4^3^ = 24^3 ; 7 and 13.

XXXIII. (Pages 245-248.)

1. x + ^y = 6', 9a;-3?/-5 = 0.

2. 25a; + 6?/ = 137; 6ic-25t/ + 20 = 0.

3. ±a;V7±4?/ = 16; ±4a;T2/V7-l V7.
5. ?/ = 3x±iVH^; (=^/^\/65, =F^VV195).

31. Use Arts. 145 and 260.

XXXIV. (Pages 262—264.)
1. x + 2y = L 2. 2a;-7'(/ + 8 = 0; (-1, -i).

3. 3a; + 8i/ = 9; 2a; = 3i/.

4. 9a;2- 24a;?/ -4^2 + 30a; + 402/ -55 = 0.

5. a2?/ + &2^ = 0; a2y-62a; = 0; ahj + h^x = 0\ ay + hx = 0.

XXXV. (Pages 268—270.)

1. x^ - 2xy cot 2a - if =a^-b'^. 2. cx^-2xy = ca^.

3. d2 (a;2 - a2)2 = 4 (62a;2 + a2|/2 - a262).

4. X (x2 - a2)2 ^ 2 (a;2?/2 + y^x^ + a22/2 - a2&2) .



X COORDINATE GEOMETKY.

5. (a;2+?/2_a2_52)2^4cot2a(&2.r2 + aV-«"&^)-

Q, ay = bxta,na. 7. &2a;2 + a2^2_4^2^2^

8. ¥x^ + aY=a^h^{a^ + h^). 9. h^x^ + aY=2a%y.

10. (62a:2 + a2l/2)2z=c2(6%2 + ^4y2).

11. (a2+&2) {b^x^ + aY)^ = a^^^{b^x^ + aY)-

12. fe^a; (a; - 7^) + ahj {y~k) = 0.

13. C2a262 (6%2 + a2j^2) + (^,2^2 _^ ^2^2 _ 1) (^)4a;2 4. ^4^2) ^ Q.

14. {b^x^ + ahjY = a^^H^^ + y^)'

15. a464 (a;2 + y2^ = (^2 + ^,2) (52^2 ^ ^2^2)2.

29. If the chords be PK and PK', let the equation to KK' be
y = mx + c ; transform the origin to P and, by means of Art. 122,
find the condition that the angle KPK' is a right angle ; substi-
tute for c in the equation to KK', and find the point of inter-
section of KK' and the normal at P. See also Art. 404.

XXXVI. (Pages 282—284.)

1. 16a;2-9i/2 = 36. 2. 25a;2- 144^2=900.

3. 65a;2-36?/2=441. 4. a;2-2/2 = 32.

5. 6, 4, {±^13, 0), 2|-. 6. 3a;2-2/2=3a2.

7. 7?/2 + 24.'c?/-24aa;-6a2/ + 15a2 = 0; (-|'«)5 12a; -9?/ + 29a =0.

8. (5, -V-)- 9. 242/-30a;=±V161.

14. 2/=±^±x/«"^^&^; {^'+^')\/^2^-

15. 9^ = 32;r. 16. 125a; -48^/ = 481.

29. (1) &%2+aV=«'&M&'-«'); (2) ^=«-5^2;

(3) a;2 (a2 + 2&2) _ a^ - 2«^^ea; + a^ {a^ - h^) = 0.

XXXVII. (Pages 295, 296.)

1. At the points (a, ±b>J2).

8. {2x + y + 2){x + 2y + l) = 0, (2a; + ?/ + 2) (.i; + 2?/ + l) = const.

9. 3a;2+ 10^;!/ + 8^/2 + 14x + 22i/ + 7 = 0;
3a;2 + lOcKy + 8i/2 + 14a: + 22i/ - 1 = 0.

XXXVIII. (Pages 302—305.)

16. (±fv/6a, =FtV6a); (^^W^^. ±x/6a).

XXXIX. (Pages 319—321.)

19. Transform the equation of the previous example to Cartesian
Coordinates.



ANSWERS. XI

XL. (Pages 331, 332.)

1. A hyperbola ; (2, 1) ; c'= - 26.

2. An ellipse; ( -|, -|); c'= -4. 3. A parabola.

4. A hyperbola ; ( - H, - -io) ; ^ = - 46.

5. Two straight lines ; (-H» if); ^' = 0.

6. A hyperbola; (-|i -gV); c'=~j\\.

7. {2x + By-l){4x-y + l) = 0; 8x^ + 10xij -Sy'^-2x + 'iy = 0.

8. {y + x-2){y-2x-S) = 0; y^-xy -2x^~5y + x + 18 = 0.

9. (lla;-2?/ + 4)(5a;-10y + 4) = 0;

55a;2 - 120x2/ + 20?/2 + 64a; - 48!/ + 32 = 0.
10. 19a;2 + 24a;?/ + 7/2-22.^-6^ + 4=0;

19a;2 + 24a;i/ + 7/2 _ 22a; - 67/ + 8 = 0.
12. a;2-7/2 = 4a2. 13. (aa;-67/)2 = (a2_^2) (^^ _ ;,a;).

14. {x-yf-2{x + y)+i = 0. 15. (.T3/ + a&)tan(a-(Q) = &.r-a7/.

16. ^2 + |^-2||cos(a-^) = sin2(a-/3). 17. A point.

18. Two straight lines. 19. A straight line and a parabola.

20. A straight line and a rectangular hyperbola.

21. A circle and a rectangular hyperbola.

22. A straight line and a circle.

23. Two imaginary straight lines.

24. A circle and a straight line. 25. A parabola.

26. A circle. 27. A hyperbola. 28. An ellipse.

XLI. (Pages 346—348.)

7 ( , ^^— ^ \ . 9. Two coincident straight lines.

'• V 676 ' 169/

10. tan^i=-|, tan^2 = fj »'i = \/3. and 7-3 =^*

11. ^1 = 45°, ^2 = 135°, ri=V2, and 7-2 = 2.

12. tan^, = 7 + 5V2; tan^2=7- 5;^2,

28. 2. 29. 5V\/3. 30. fV^-

31. (TyVVioTi, ^i^^VVio^); hJ^oT^io.

fa a ,„ 3rt , a ,_\ , ,„

32. (2-^4'^^' T'=2^^j' ^^^•

33. (-|=fW6, 4±iV6); K/3.

34. ( - l±lv/6, l±tv/6); 2.



Xll COORDINATE GEOMETKY.

XLII. (Pages 354, 355.)

1. (1) 3; (2) 3; (3) 4; (4) 2; (5) 4; (6) 3; (7) 3.

10. Ax + Hy = and Hx + By = 0; II^ = AB, so that the conic is a
pair of parallel straight lines.

11. .r(.c + 3?/) = 0; (2a;-3?/)2=0.

XLIII. (Pages 363, 364.)

1. A conic touching 8 = where T=0 touches it and having its
asymptotes parallel to those of *S = 0.

A conic such that the two parallel straight lines u=Q and
?t + A; = pass through its intersections with 5 = 0.

XLIV. (Pages 375—377.)

6. (-1,5) and (4, -3). 7. (-|,-f). 8. (^^ ' ^) •

9. (-4, -4) and (-1, -1); x-\-y + l = and a; + i/ + 3 = 0.

15. If P be the given point, G the centre of the given director circle,
and PCP' a diameter, the focus S is such that PS.P'S is
constant.

16. If PP' he the given diameter and S a focus then PS.P'S is
constant.

XLV. (Pages 383, 384.)

5. Qx^ + 12xy + ly^-12x-lSy = 0.

17. The narrow ellipse (Art. 408), which is very nearly coincident with
the straight Hne BD, is one of the conies inscribed in the quadri-
lateral, and its centre is the middle point of BD. This middle
point, and similarly the middle points of -4C and OL, therefore
lie on the centre-locus.

XLVI. (Pages 390—392.)

7. Proceed as in Art. 413, and use, in addition, the second result
of Art. 412, Cor. 2. From the two results, thus obtained,
eliminate 8.

9. Take l^^x + vi^y -1 = (Art. 412, Cor. 1) as a focal chord of the
ellipse.

14. If the normals are perpendicular, so also are the tangents ; the
line IjX + nijy — 1 = is therefore the iDolar with respect to the

ellipse of a point {sja^ + b'^ cos,6, sja^ + b'-' sin.6) on the director
circle.

15. The triangle ABC is a maximum triangle (Page 235, Ex. 15)
inscribed in the ellipse.

20. Use the notation of Art. 333.



ANSWERS. Xlll



XL VII. (Pages 397, 398.)

11, The locus can be shewn to be a straight line which is perpendi-
cular 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.



XL VIII. (Pages 405—407.)

5. Two of the normals drawn from coincide, since it is a centre of
curvature. The straight line l■^x + 7n-j^^tJ = l (Art. 412) is therefore
a tangent to the ellipse at some point <p and hence, by Art. 412,
the equation to QB can be found in terms of (p.



XLIX. (Pages 414—416.)



c-



1. {by- ax- cy^='iacx. 2. x^ + y'^- c {x + y) + - = 0.

x^ w^
3. -^ + ^2=1. 4. A parabola touching each of the two lines.

5. A central conic. 6. A parabola. 7. a^x^ + h-y^ = c'^.

19. The line joining the foci is a particular case of the confocals and
the polar of O with respect to it is the major axis ; the minor
axis is another particular case, so that two of the polars are lines
through G at right angles ; also the tangents at to the con-
focals through it are two of the polars, and these are at right
angles. Thus both G and are on the directrix.

21. The crease is clearly the line bisecting at right angles the line
joining the initial position of G to the position which C occupies
when the paper is folded.

__ Zcosa ^ -

23. =1 -e cosa cos^.



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