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circles of curvature at which pass through a given point O, not on the

ellipse. If be on the ellipse, why is the number of circles of

curvature passing through it only four?

8. The circles of curvature at three points of an ellipse meet in a

point P on the curve. Prove that (1) the normals at these three

points meet on the normal drawn at the other end of the diameter

through P, and (2) the locus of these points of intersection for

different positions of P is the ellipse

4(a%2 + &2^2)^(^2_62)2,

9. Prove that the equation to the circle of curvature at any point

{x', y') of the rectangular hyperbola x^-y^ = a^ is

a^ (a;2 + y^)- ^xx'^ + iyy'^ + Sa^ (a;'2 + y'^) = 0.

10. Shew that the equation to the chord of curvature of the

rectangular hyperbola xy = c^ at the point "Â«" is ty + t^x = c{l + t'^),

and that the centre of curvature is the point

V^-2^' '^'

Prove also that the locus of the .pole of the chord of curvature is

the curve r^ = 2c^ sin 29.

11. PQ is the normal at any point of a rectangular hyperbola and

meets the curve again in Q ; the diameter through Q meets the curve

again in R ; shew that FR is the chord of curvature at P, and that

PQ is equal to the diameter of curvature at P.

12. Prove that the equation to the circle of curvature of the conic

ax^+'2hxy + by^=i2y at the origin is

a{x'^ + y^) = 2y.

13. If two confocal conies intersect, prove that the centre of

curvature of either curve at a point of intersection is the pole of the

tangent at that point with regard to the other curve.

XLVIIIJ ENVELOPES. 407

14. Shew that the equation to the parabola, having contact of the

third order with the rectangular hyperbola icy = c^ at the point

(-. i)-

is {x-yt^)^-4ct{x + yt^) + 8cH^ = 0.

Prove also that its directrix bisects, and is perpendicular to, the

radius vector of the hyperbola from the centre to the point of contact.

15. ^ Prove that the equation to the parabola, which passes through

the origin and has contact of the second order with the parabola

y'^â€”^ax at the point (at^^ 2ai), is

(4a; - %tyf + 4aÂ«2 (3a; - "Ity) = 0.

16. Prove that the equation to the rectangular hyperbola, having

contact of the third order with the parabola y^ = 4cax at the point

{at^^ 2af), is

x^ - 2txy - 2/2 + 2aa; (2 + 3Â«2) - 2at^y + a^t^ = 0.

Prove also that the locus of the centres of these hyperbolas is an

equal parabola having the same axis and directrix as the original

parabola.

17. Through every point of a circle is drawn the rectangular

hyperbola of closest contact; prove that the centres of all these

hyperbolas lie on a concentric circle of twice its radius.

18. A rectangular hyperbola is drawn to have contact of the third

X^ 7/2

order with the ellipse -^ + ^2 = ^ 5 ^^^ i*^ equation and prove that the

locus of its centre is the curve

/a;2 + i/2y

"^ ^ "^ &2 '

Envelopes.

430. Consider any point P on a circle whose centre

is and whose radius is a. The straight line through P

at right angles to OP is a tangent to the circle at P.

Conversely, if through we draw any straight line OP oi

length a, and if through the end P we draw a straight

line perpendicular to OP^ this latter straight line touches,

or envelopes, a circle of radius a and centre 0, and this

circle is said to be the envelope of the straight lines drawn

in this manner.

Again, if S be the focus of a parabola, and PY be the

tangent at any point P of it meeting the tangent at the

408 COORDINATE GEOMETRY.

vertex in the point Y, then we know (Art. 211, 8) that

STP is a right angle. Conversely, if S be joined to any

point J' on a given line, and a straight line be drawn

through Y perpendicular to SY, this line, so drawn, always

touches, or envelopes, a parabola whose focus is S and such

that the given line is the tangent at its vertex.

431. Envelope. Def. The curve which is touched

by each of a series of lines, which are all drawn to satisfy

some given condition, is called the Envelope of these

lines.

As an example, consider the series of straight lines

which are drawn so that each of them cuts off from a pair

of fixed straight lines a triangle of constant area.

We know (Art. 330) that any tangent to a hyperbola

always cuts off a triangle of constant area from its asymp-

totes.

Conversely, we conclude that, if a variable straight line

cut off a constant area from two given straight lines, it

always touches a hyperbola whose asymptotes are the two

given straight lines, i. e. that its envelope is a hyperbola.

432. If the equatio7i to any curve involve a variable

parameter, in the first degree only, the curve always passes

through a fixed point or points.

For if X be the variable parameter, the equation to the

curve can be written in the form S + \S' â€” 0, and this

equation is always satisfied by the points which satisfy

S = and S' = 0, i.e. the curve always passes through the

point, or points, of intersection oi S-Q and S' = [compare

Art. 97].

433. Curve touched hy a variable straight line whose

equation involves, in the second degree, a variable parameter.

As an example, let us find the envelope of the straight

lines given by the equation

m^x â€” my + a = (1),

where m is a quantity which, by its variation, gives the

series of straight lines.

ENVELOPES. 409

If (1) pass through the fixed point (A, ^), we have

w^h â€” mk + a = (2).

This is an equation giving the values of m correspond-

ing to the straight lines of the series which pass through

the point (A, li). There can therefore be drawn two

straight lines from (A, ^) to touch the required envelope.

As (A, Aj) moves nearer and nearer to the required

envelope these two tangents approach more and more

nearly to coincidence, until, when (A, h) is taken on the

envelope, the two tangents coincide.

Conversely, if the two tangents given by (2) coincide,

the point (A, U) lies on the envelope.

Now the roots of (2) are equal if l? = 4:ah,

so that the locus of (A, k), i. e. the required envelope, is the

parabola y^ = 4:ax.

Hence, more simply, the envelope of the straight line (1 )

is the curve whose equation is obtained by writing down

the condition that the equation (1), considered as a quad-

ratic equation in m, may have equal roots.

By writing (1) in the form

a

y = mx H â€” ,

m

it is clear that it always touches the parabola y^ = ^ax.

In the next article we shall apply this method to the

general case.

434. To find the envelope of a straight line whose

equation involves^ in the second degree, a variable parameter.

The equation to the straight line is of the form

X^P-{-\Q + R = (1),

where X is a variable parameter and P, Q^ and M are

expressions of the first degree in x and y.

Equation (1) may be looked upon as an equation

giving the two values of X corresponding to any given

point T.

410 COORDINATE GEOMETRY.

Through this given point two straight lines to touch the

required envelope may therefore be drawn.

If the point T be taken on the required envelope, the

two tangents that can be drawn from it coalesce into the

one tangent at T to the envelope.

Conversely, if the two straight lines given by (1)

coincide, the resulting condition will give us the equation

to the envelope.

But the condition that (1) shall have equal roots is

Q^ = 4.PR (2).

This is therefore the equation to the required envelope.

Since P, Q, and R are all expressions of the first degree,

the equation (2) is, in general, an equation of the second

degree, and hence, in general, represents a conic section.

The envelope of any straight line, whose equation

contains an arbitrary parameter and square thereof, is

therefore always a conic.

435. The method of the previous article holds even if

P, Q, and R be not necessarily linear expressions. It

follows that the envelope of any family of curves, whose

equation contains a variable parameter X, in the second

degree, is found by writing down the condition that the

equation, considered as an equation in X, may have equal

roots.

436. Ex. 1. Find the envelope of the straight line which cuts off

from two given straight lines a triangle of constant area.

Let the given straight lines be taken as the axes of coordinates and

let them be inclined at an angle w.

The equation to a straight line cutting off intercepts / and g from

the axes is

rl=' <^>-

If the area of the triangle cut off be constant, we have

\f .g . sin w = const.,

i.e, fg=zconst.=K^ (2).

On substitution for g in (1), the equation to the straight line

becomes f'y -fIO + K^x=:0.

ENVELOPES. EXAMPLES. 411

By the last article, the envelope of this line, for different values of

/, is given by the equation

I.e. xy=-^.

The result is therefore a hyperbola whose asymptotes coincide with

the axes of coordinates.

Ex. 2. Find the envelope of the straight line ivhich is such that

the product of the perpendiculars draiun to it from two fixed points is

constant.

Take the middle point of the line joining the two fixed points as

the origin, the line joining them as the axis of x, and let the two

points be {d, 0) and {-d, 0).

Let the variable straight line have as equation

y=mx + c.

The condition then gives

md + c -md + c , , ^â€ž

X = constant = ^%

jjl + m^ Jl + m^'

so that c^ - viH^ =A^{1+ m^) .

The equation to the variable straight line is then

y-'mx=c= fJ{A'^ + d^)m^ + A^.

Or, on squaring,

mP (a;2 -A^- d^) - 2mxy + (y^ - A^) = 0.

By Art. 435, the envelope of this is

[2xyf = 4 (a;2 - ^2 _ ^2) (^2 _ ^2) ^

This is an ellipse whose axes are the axes of coordinates and whose

foci are the two given points.

Ex. 3. Find the envelope of chords of an ellipse the tangents at the

end of ivhich intersect at right angles.

Let the ellipse be â€” â€ž + ^r, = 1.

a^ 0^

If the tangents intersect at right angles, their point of intersection

P must lie on the director circle, and hence its coordinates must be of

the form (c cos ^, c sin 6), where c = Ja'^+ h^.

The chord is then the polar of P with respect to the ellipse, and

hence its equation is

x . c cos 6 V .c sin d ^

412 COORDINATE GEOMETRY.

a

Let t^tan - . Then since

l-*an2- ^_^2 2(

1 + tan^ -

the equation to the line is

ex 1-t'^ cy 2t _^

The envelope of this is (Art. 434),

^2g2 ^2g2

^2 2/^ _i

Since -:; â€” r-, s â€” r^ = a^ - &2 this equation represents a conic

a^ + b^ a^ + b^

confocal with the given one.

Ex. 4. The normals at four points of an ellipse meet in a point ;

if the line joining one pair of these points pass through a fixed point,

prove that the line joining the other pair envelopes a parabola which

touches the axes.

Let the equation to the ellipse be

%*t=^ W'

and let the equation to the two pairs of lines through the points be

lx+m7j = l (2),

and lj^x + miy = l (3).

By Art. 412, Cor. (1), we then have

11-^= â€” 2 ^^^ *^%=~i^ W*

If the straight line (3) pass through the fixed point (/, g), we have

BO that, by (4), -X_^|_=l,

and therefore 1= â€” ^ -r^ .

ENVELOPES. EXAMPLES. 413

If this value of I be substituted in (2), it becomes

m^a%^y + m {a?gy - W-fx - a^ft^) - d^g = 0,

the envelope of which is

(a^gy - hjx - a%Y = - ^a?g . a^&s^ ,

i.e, {a^gy - V^fxf + 2.a%^ {bjx + a?gy) + a%'^=0 (5) .

This is a parabola since the terms of the second degree form a

perfect square. Also, putting in succession x and y equal to zero, we

get perfect squares, so that the parabola touches both axes.

437. To find the envelope of the straight line

lx + TYiy + n â€” ^ (1),

where the quantities I, m, and n are connected hy the

relation

aP + hm? + cn^ + Ifmn + "Ignl + Vilm = (2).

[Equation (1) contains two independent parameters â€”

and - , whilst (2) is an equation connecting them. We

n

could therefore solve (2) to give - in terms of â€” : on sub-

^ ' n n

stituting in (1) we should then have an equation containing

one independent parameter and its envelope could then be

found.

It is easier, however, to proceed as follows.]

Eliminating n between (1) and (2), we see that the

equation to the straight line may be written in the form

aV' + htn^ â– \-g(}x-\- myY â€” 2 {fm + gl) (Ix + my) + 2hlm = 0,

/ ^\^ I

i. e. (a â€” 2gx + cx^) ( â€” j + 2 {cxy â€” gy -fx + h) â€”

^{h-2fy + cf) = 0.

The envelope of this is, by Art. 435,

{cxy - gy -fx + hf = {a- 2gx + ex") (b - 2fy + cy\

i.e., on reduction,

x^ (be -D + y' (ca - g') + 2xy (fg - ch)

+ 2x{fh-bg)^-2y{gh-af) + ab-h'' = 0.

414 COORDINATE GEOMETRY.

The envelope is therefore a conic section.

Cor. The envelope is a parabola if

i. e. if c = 0, or if abc + %fgh â€” af^ â€” hg^ - ch^ ~ 0.

438. Ex. Find the envelope of all chords of the parabola y^ = 4aa;

lohich subtend a given angle a at the vertex.

Any straight line is

Za3 + m7/ + w = (1).

The lines joining the origin to its intersections with the parabola

are, (by Art. 122), ny^ = - 4:ax {Ix + my) ,

i.e. ny^ + 4:a'mxy + 4alx^=0.

If a be the angle between these lines, we have

2/J'ia^m^-4aln

tana=â€” ^^ j-z ,

n+4:al

i.e. 16a2Z2 - IQa^ cot^ am^+n^ + 8aln{l + 2 cot^ a) = 0.

With this condition the envelope of (1) is, by the last article,

a;2 ( - 16a2 cot2 a) + 2/2 [I6a2 _ (4^ + 8a cot2 a)2]

+ 2x . 16a2 cot2 a (4a + 8a Cot2 a) - 256a4 cot2 a= 0,

i.e. the ellipse

[a; - 4a (1 + 2 cot2 a)]2 + 4 cosec2 a . 2/2= 64 cot2 a . cosec2 a.

EXAMPLES. XLIX.

Find the envelope of the straight line - + ^=1 when

a p

1. aa + b^=c. 2. a + ^+s/aF+^=c.

b^ a^

^' ^2 + ^2=1-

Find the envelope of a straight hne which moves so that

4. the sum of the intercepts made by it on two given straight

lines is constant.

5. the sum of the squares of the perpendiculars drawn to it from

two given points is constant.

6. the difference of these squares is constant.

7. Find the envelope of the straight line whose equation is

ax cos 9 + by sin 6 = c^.

[EXS. XLIX.] ENVELOPES. EXAMPLES. 415

8. Circles are described touching each of two given straight lines ;

prove that the polars of a given point with respect to these circles all

touch a parabola.

9. From any point P on a parabola perpendiculars P3I and FN

are drawn to the axis and tangent at the vertex; prove that the

envelope of MN is another parabola.

10. Shew that the envelope of the chord which is common to the

parabola y^=:4:ax and its circle of curvature is the parabola

y^ + 12ax = 0.

11. Perpendiculars are drawn to the tangents to the parabola

y^=4:ax at the points where they meet the straight line x = b; prove

that they envelope another parabola having the same focus.

12. A variable tangent to a given parabola cuts a fixed tangent in

the point A ; prove that the envelope of the straight line through A

perpendicular to the variable tangent is another parabola.

13. Shew that the envelope of chords of a parabola the tangents

at the ends of which meet at a constant angle is, in general an ellipse.

14. A given parabola slides between two axes at right angles ;

prove that the envelope of its latus rectum is a fixed circle.

15. Prove that the envelope of chords of an ellipse which subtend

a right angle at its centre is a concentric circle.

16. If the lines joining any point P on an ellipse to the foci meet

the curve again in Q and R, prove that the envelope of the line QE is

the concentric and coaxal ellipse

x^ y^ fl + ey_

17. Prove that the envelope of chords of the rectangular hyperbola

xy=a^, which subtend a constant angle a at the point (a?', y') on the

curve, is the hyperbola

x^x'^ + yh^'^=2a^xy (1 + 2 cot^ a) - 4a^ cosec^ a.

18. Chords of a conic are drawn subtending a right angle at a

fixed point 0. Prove that their envelope is a conic whose focus is

and whose directrix is the polar of with respect to the original conic.

19. Shew that the envelope of the polars of a fixed point with

respect to a system of confocal conies, whose centre is 0, is a parabola

having GO as directrix.

20. A given straight line meets one of a system of confocal conies

in P and Q, and BS is the line joining the feet of the other two

normals drawn from the point of intersection of the normals at P and

Q ; prove that the envelope of RS is a parabola touching the axes.

416 COORDINATE GEOMETRY. [ExS. XLIX.]

21. ABGD is a rectangular sheet of paper, and it is folded over so

that G lies on the side AB ; prove that the envelope of the crease so

formed is a parabola, whose focus is the initial position of G.

22. A circle, whose centre is A, is traced on a sheet of paper and

any point B is taken on the paper. If the paper be folded so that the

circumference of the circle passes through B, prove that the envelope

of the crease so formed is a conic whose foci are A and B.

23. In the conic -=l-ecos^ find the envelope of chords which

r

subtend a constant angle 2a at the focus.

24. Circles are described on chords of the parabola y^ = 'iax, which

are parallel to the straight line lx + my = 0, as diameters; prove that

they envelope the parabola

[ly + 2maf = 4a (Z^ + m^) {x + a).

25. Prove that the envelope of the polar of any point on the circle

{x-\-af+{y + hf=h'^ with respect to the circle x^+y^=c^ is the conic

h^ (a;2 + 2/2) = [ax + hy + c^-

26. Chords of the conic -=l-ecos^ are drawn passing through

r

the origin and on these circles as diameters circles are described.

Shew that the envelope of these circles is the two circles

- â€” h ecoBd ) = lÂ±e.

r \r J

ANSWERS.,

I. (Pages 14, 15.)

1. 5. 2. 13. 3. 3V7. 4. s/aJ+b\

5. ^a2 + 2&2 + c2-2afc-26c. 6. 2a sin -^^.

7. a {j)x^ - m^) sj{m^ + m.^f + 4. 9. 3Â±2^15.

15. (Â¥->Â¥)â€¢ 16. (-2,-9). 17. (1, -I); (-11, 16).

18. (-5H, 2A); (-20i, 34|). 19. (-i,0); (-1, 2).

20. (-1,1); (1.1); (I, -I).

^^- \'a + b' ^+& 7' VÂ«-^ ' a-bj'

^^- \^ /c + Z + m ' h + l + m J'

II.' (Pages 18, 19.)

1. 10. 2. 1. 3. 29. 4. 2ac.

5. a?. 6. 2fl& sin ^IZ,^? gin ^^I^^ sin ^LlL^a .

i Jj a

7. a2(w2-Â»n3)(m3-mi)(mi-W2).

8. la^im^-m^im^-m-^ivi-^-m^.

9 . la'^ (mg - W3) (wig - wij) (m^ - mg) -f- mjin^m^ .

13. 201. 14, 96.

III. (Pages 22, 23.)

12. 2^5. 13. x/79. 14. s/7a. 16. 1(8-3^3).

17. '^-. 18. iÂ«V3- 25. r^=ci\ 26. ^=a.

27. r=2acose. 28. r cos 2^ = 2a sin ^. 29. r cos ^ = 2a sin^ ^.

30. r2 = a2 cos 2^. 31. x^ + 2f = a^. 32. y = mx.

33. a;2 + 2/2 = aa;. 34. {x^ + 2ff = 4:a^xY-

35. (a;2 + 2/2)2z=a2(a;2-2/2). 35, a;i/ = a2. 37. x'^-if = a\

38. 2/2 + 4flx = 4a2. 39. 2/2 = 4aa; + 4a2.

40. ^^ - 3x^2 ^ 3a;22^ -y^= 5kxy.

L. 27

11 COORDINATE GEOMETRY.

IV. (Page 30.)

8. 2ax + k^-=^0. 9. Oi2-l)(a;2 + ?/+a2) + 2aa;{w2 + l) = 0.

10. 4a;2(c2-4a2) + 4cV = c2(c2-4a2). H. {Qa-2c) x = a^-c\

12. ^f-i^J-2x + 5 = 0. 13. ^y + 2x + S = 0. 14. x + y = l.

15. y = x. 16. ?/ = 3.r. 17. 15a;2-2/2 + 2aa; = a2.

18. .r2 + 2/2 = 3. 19. fl^2 + ^2^4y^

20. 8a:2 + 8?/2 + 6;r-36;c + 27=:0. 21. .'c2 = 3?/2.

22. X'-\-2ay = a'^.

23. (1) 4a;2+3?/2 + 2a2/ = a2; (2) x'' -^if + Say = 4.a^.

V. (Pages 41, 42.)

1. y=x + l. 2. a;-2/-5 = 0. 3. x-y >^^-2^2> = 0.

4. 52/-3a; + 9=:0. 5. 2a; + 3y = 6. 6. 6a; - 5?/ + 30 = 0.

7. (1) a; + 2/ = ll; (2) 2/-a;=l. 8. x^y + 1 = ; x-y = ^.

9. a;^' + a:V = 2.-cy. 10. 20?/ -9a; = 96. 15. x-\-y=0.

16. 2/-^ = l- 17. ly + 10x = ll.

18. ax - by = (lb. 19. (Â«-2&) a;- &?/ + &^ + 2a&- a2 = 0.

20. ^j(Â«i + i2)-2a;=:2fl^if2. 21. Â«ii22/ + ^ = Â« {*i + ^2)-

22. X cos 1 (01 + 02) + 2/ sin i (01 + 02) = a cps J (01 - 02).

Â«Â« ^ 01 + 02 2/ â€¢ 01 + 02 01 ~ 02

23. -cos?^+|sin^^=cos'^-^ - ^.

24. &a; cos 4 (01 - 02) - Â«2/ sin J (0i + 02) = ab cos i (0i + 02).

25. x + Sy + 7 = 0; y-3x=l; y + 7x = ll.

26. 2x-Sy = 4:; y-Sx=l; x + 2y = 2.

27. 1/ (a' -a)-x {V -b) = a'b - ah' ; y {a' - a) J^x{b'-b) = a'b' - ab.

28. 2ay-2b'x = ab-a'b'. 29. y = ^x\ 2?/ = 3x.

VI. (Pages 48, 49.)

1. 90Â°. 2. tan-iff. 3. tan-i-f. 4. 60Â°.

8. 4y + 3a; = 18. 9. 7?/-8rc = 118. 10. 4i/ + lla;=10.

11. a; + 4?/ + 16 = 0. 12. a.r + ftr/ = a\

13. 2a; (a - a') + 2?/ (6 - &') = a^ - a"^ + 62- y^.

15. yx' -xy' = 0; a^xy' -b^x'y = {a^-b^)x'y' ; xx' -yy' = x' - y'^.

16. 121y-88a; = 371; 33?/ -24a; =1043.

17. a; = 3; 2/ = 4; 4i. 19. a; = ; 2/ + x/3a; = 0.

20. 2/ = ^; (l-m2)(?/-7f) = 2m(a;-/i).

21. tan-iH; 9a;-7i/ = l; 7a; + 9?/ = 73.

ANSWEES. m

VII. (Pages 53, 54.)

1. ^. 2. 2f. 3. 5^. 4.

5. a cos J (a -/S). 8.

a^ + ab-b^

Ja^ + b'-^

^1 + m^

9. |^(&Â±V^MT^), ol. 11. 4(2 + ^3).

VIII. (Pages 61-65.)

/ - 11 41\ _a&_ a& \

â– 'â– â€¢ V 29" '297* a + 6 'Â« + &;â€¢

,4. {a cos i (01 + 02) sec i (01 -0o), asini (01 + 0^)860 i(0i- 0^)}.

/ a{b-b') 2bb' \ 130

^" \ b + b' ' b + b'J' ^' 17^29'

8. y = a', %y = 4.x + ^a. 9. (1,1); 45Â°.

10. (f,i); tan-160. 11. (-1, -3); (3,1); (5,3).

12. (2, 1); tan-i,^. 13. 45Â°; (-5, 3) ; a;- 3^ = 9; 2.r-?/ = 8.

14. 3 and - f . 19. w?! (Â«2 ~ ^3) + ^% (^3 ~ '^1) + "^3 (^1 ~ ^2) = 0-

20. (-4,-3). 21. (h/-!^)- ^ 23. 43.i;-29y = 71.

24. x-y=ii. 25. 2/ = 3jc. 26. y = x.

27. a^y-b^x=ab{a-b). 28. 3;r + 42/ = 5a. 29. a; + ?/ + 2 = 0.

30. 23a; + 23^ = 11. 31. 13.r - 23?/ = 64.

33. ^ic + 5j/ + + X (^'a; + J5 '?/ + C) = where A is

(1) -^,, (2) -;g7, (3) -^, - ^, and (4) - ^.^.^^y^ ^..

37. y = 2', x = 6. 38. 99a; + 77?/ + 71 = 0; 7a;-9y-37=0.

39. x-2y + l = 0; 2x + y=:S.

40. ^(2V2-3)+?/(x/2-l) = 4V2-5;

a;(2V2 + 3) + ?/(V2 + l) = 4^2 + 5.

41. {y-b){m + m') + {x-a){l-mm') = 0;

[y - 6) (1 - ?u7?i') -{x- a) (m + ?/t') = 0.

42. 33a; + 9?/ = 31; 112ic- 64?/ + 141 = 0; 7y-x = 18.

43. a;(3+V17)+2/(5 + V17) = 15 + 4^17;

a;(4 + V10) + ?/(2 + ^10) = 4VlO + 12;

a; (2 V34 - 3 V5) +2/ ( V34 - 5V5) = 6 V34 - 5 V5.

44. A{y-k)-B{x-h)=^{Ax + By + C).

45. At an angle of 15Â° or 75Â° to the axis of x.

27â€”2

IV COORDINATE GEOMETRY.

IX. (Pages 72, 73.)

I. (1) tan-i^; (2) 15Â°. 2. tan"!^^.

3. tan-i^^^tancj.

7, y = 0, y = x-a, x = 2a, y = 2a, y = x + a, x = 0, y = x, x = a, and

y = a, where a is the length of a side.

10. 2/(6-V3) + ^(3V3-2) = 22-9V3. 11. h

12. 10?^ - 11a; + 1 = ; ^^ ^IH-

X. (Pages 78â€”80.)

4. (-7,3). 5. (-ii, If); m-

/-85 + 7V5 7V5-27 \ 35-7^5 â€ž n^.^.^.

^- \ 120"""' W^'J' 120" â€¢ 'â€¢ ^^'S'^*'"^?-

(6 + ^/10 2 + VlO ] / 6-^10 2-V10\ /8-VlO lo + ^lO N

Â°- I 2 ' 2 j'V 2 ' 2"y'V 6 ' 6 J'

9. {%, f), (2, 12), (12, 2), and ( - 3, - 3) ; 1^2, 4^2, 4^2, and 6^2.

10. (-13^,194). 11. 4. 12. 7||. 13. f.

14. ^. 15. M&-c)(c-a)(a-&).

16. Â«^ (w^a - ^'^3) {>^h - ''^h) (^'h " 'm2)^2m^m^m^.

,Â« lf(C2-Cs)^ (Cs-Ci)2 (Ci-Co)2]

17. i(Ci-Co)2-f K-wi2- 18. oV-^ ^ + i-3 iL + U 2^1

23. (f,f)-

24. 10i/ + 32Â« + 43 = 0; 25a; + 29?/ + 5 = 0; 2/ = 5a; + 2; 52a; + 80i/ = 47.

26. (4 + iv/3, f + v/3) ; (4 + i^/3, f + W3).

XI. (Pages 85â€”87.)

1. a;2 + 2a;?/cota-2/2 = a2, 2. ?/2 + Xa;2=:Xa2.

3. (7?i + l)a;=(m-l)a. 4. (w + ?i) (a;2 + ?/'-' + a^)- 2aa;(m-/Â«) = c2.

5. .T + 2/ = csec2|. 6. a;-?/ = <icosec2-.

7, j; + ?/ = 2ccosecw. 8. 2/-^ = 2ccosecw,

9. a:2 + 2a;i/cosw + 2/^=4c2cosec2w.

10. (^^ + y^) cos w + a;?/ (1 + cos^ w) = a; [a cos w + &) + 2/ (& cos w + a).

II. a;(m + cosw)+i/(l+?Â«cosw) = 0.

12. (i) ai/2 + &a;2+(a + &)ic?/-a?/(a + 2&)-&a;(2rt + 6) + a&(a + 6) = 0;

(ii) y=a;. 19. A straight line.

20. A circle, centre 0. 25. A straight line.

27. If P he the point [h, k), the equation to the locus of S is

h k ,

- + - = 1.

a; 2/

ANSWERS. V

XII. (Page 94.)

1. {x-Sy){x-'iy) = 0;ta.n-'^^\. 2. {2.t- ll?/)(2a;-?/) = 0; tan-i -J.

3. (lla; + 2y)(3a;-7i/)=0; tan-iff. 4. x = l ; x = 2', x = 3.

5. 2/= Â±4. 6. {y + ^x){y-2x){y-Sx) = 0;ia.n-^{-^);t&n-^{}).

7. a;(l-sin^)+?/cos^ = 0; a; {l + sin^) + ^cos^ = 0; 0.

8. r/siii^ + a;cos^= Â±a;^/cos2^; tan-^ (cosec^ Aycos2^).

9. 12x^-7xy-12y^ = 0; Ux^ + Hxy -71y^ = 0\ x^-y"^0',

x^-y^=0.

XIII. (Pages 98, 99.)

1, (I, -Â¥);45Â«. 2. (2, 1); tan-if. 3. (-f, -t);90Â°.

4. (-1, 1); tan-13. 6. -15. 7. 2. 8. -10 or -171.

9. -12. 10. 6. 11. 6. 12. 14. 13. -3.

14. fori=f. 16. (i) c{a + h) = 0; {ii) e = 0, ox ae = bd.

17. Siz + e.'c^se; 5?/-6a;=14.

XV. (Page 112.)

1. (1) i/'2=4a;'; (2) 2x'^ + y"' = 6.

2. (1) a;'2 + 2/'2=:2ca;'; (2) x'^ + y'^=2cy\

3. (a-&)2(a;'2 + ^'2) = a262.

4. (1) 2x'y' + a^=0; 9x'^ + 25y'^ = 225', x'^ + y'^ = l.

5. x"^ + 2j"^ = r^', x'^-y'^=a^cos2a. 6. a;'^ - 4?/'2 = ^2.

8. tan-ij; -C-^^^M^a,

XVI. (Page 117.)

1. 2x'-^Qy' + l=::0. 2. a;'2 + ^3a;y = l. 3. x'^ + y'^-=8.

4. ?/'2=4a;'cosec2a.

XVII. (Pages 123â€”125.)

I x^ + y^ + 2x-4:y = L 2. a;2 + ?/2+10a; + 12?/ = 39.

3^ x^ + y^-2ax + 2hy = 2ab. 4. a;^ + 7/2 + 2fla; + 26i/ + 262 = 0.

5. (2,4); ^61. 6. (f, l);ix/l3. 7. (^ o) 5 ^ ^-

8. {o,-f);Jp+g'- 9. (-7 - ^. -7tT=0 ' '"

13. 15a;2 + 15?/2- 94.^ + 18^ + 55=0.

14. &(a;2 + 2/2-a2) = a;(&2 + ;i2_^2), ^5^ a:2 + 7/ - aa; - fcy = 0.

ellipse. If be on the ellipse, why is the number of circles of

curvature passing through it only four?

8. The circles of curvature at three points of an ellipse meet in a

point P on the curve. Prove that (1) the normals at these three

points meet on the normal drawn at the other end of the diameter

through P, and (2) the locus of these points of intersection for

different positions of P is the ellipse

4(a%2 + &2^2)^(^2_62)2,

9. Prove that the equation to the circle of curvature at any point

{x', y') of the rectangular hyperbola x^-y^ = a^ is

a^ (a;2 + y^)- ^xx'^ + iyy'^ + Sa^ (a;'2 + y'^) = 0.

10. Shew that the equation to the chord of curvature of the

rectangular hyperbola xy = c^ at the point "Â«" is ty + t^x = c{l + t'^),

and that the centre of curvature is the point

V^-2^' '^'

Prove also that the locus of the .pole of the chord of curvature is

the curve r^ = 2c^ sin 29.

11. PQ is the normal at any point of a rectangular hyperbola and

meets the curve again in Q ; the diameter through Q meets the curve

again in R ; shew that FR is the chord of curvature at P, and that

PQ is equal to the diameter of curvature at P.

12. Prove that the equation to the circle of curvature of the conic

ax^+'2hxy + by^=i2y at the origin is

a{x'^ + y^) = 2y.

13. If two confocal conies intersect, prove that the centre of

curvature of either curve at a point of intersection is the pole of the

tangent at that point with regard to the other curve.

XLVIIIJ ENVELOPES. 407

14. Shew that the equation to the parabola, having contact of the

third order with the rectangular hyperbola icy = c^ at the point

(-. i)-

is {x-yt^)^-4ct{x + yt^) + 8cH^ = 0.

Prove also that its directrix bisects, and is perpendicular to, the

radius vector of the hyperbola from the centre to the point of contact.

15. ^ Prove that the equation to the parabola, which passes through

the origin and has contact of the second order with the parabola

y'^â€”^ax at the point (at^^ 2ai), is

(4a; - %tyf + 4aÂ«2 (3a; - "Ity) = 0.

16. Prove that the equation to the rectangular hyperbola, having

contact of the third order with the parabola y^ = 4cax at the point

{at^^ 2af), is

x^ - 2txy - 2/2 + 2aa; (2 + 3Â«2) - 2at^y + a^t^ = 0.

Prove also that the locus of the centres of these hyperbolas is an

equal parabola having the same axis and directrix as the original

parabola.

17. Through every point of a circle is drawn the rectangular

hyperbola of closest contact; prove that the centres of all these

hyperbolas lie on a concentric circle of twice its radius.

18. A rectangular hyperbola is drawn to have contact of the third

X^ 7/2

order with the ellipse -^ + ^2 = ^ 5 ^^^ i*^ equation and prove that the

locus of its centre is the curve

/a;2 + i/2y

"^ ^ "^ &2 '

Envelopes.

430. Consider any point P on a circle whose centre

is and whose radius is a. The straight line through P

at right angles to OP is a tangent to the circle at P.

Conversely, if through we draw any straight line OP oi

length a, and if through the end P we draw a straight

line perpendicular to OP^ this latter straight line touches,

or envelopes, a circle of radius a and centre 0, and this

circle is said to be the envelope of the straight lines drawn

in this manner.

Again, if S be the focus of a parabola, and PY be the

tangent at any point P of it meeting the tangent at the

408 COORDINATE GEOMETRY.

vertex in the point Y, then we know (Art. 211, 8) that

STP is a right angle. Conversely, if S be joined to any

point J' on a given line, and a straight line be drawn

through Y perpendicular to SY, this line, so drawn, always

touches, or envelopes, a parabola whose focus is S and such

that the given line is the tangent at its vertex.

431. Envelope. Def. The curve which is touched

by each of a series of lines, which are all drawn to satisfy

some given condition, is called the Envelope of these

lines.

As an example, consider the series of straight lines

which are drawn so that each of them cuts off from a pair

of fixed straight lines a triangle of constant area.

We know (Art. 330) that any tangent to a hyperbola

always cuts off a triangle of constant area from its asymp-

totes.

Conversely, we conclude that, if a variable straight line

cut off a constant area from two given straight lines, it

always touches a hyperbola whose asymptotes are the two

given straight lines, i. e. that its envelope is a hyperbola.

432. If the equatio7i to any curve involve a variable

parameter, in the first degree only, the curve always passes

through a fixed point or points.

For if X be the variable parameter, the equation to the

curve can be written in the form S + \S' â€” 0, and this

equation is always satisfied by the points which satisfy

S = and S' = 0, i.e. the curve always passes through the

point, or points, of intersection oi S-Q and S' = [compare

Art. 97].

433. Curve touched hy a variable straight line whose

equation involves, in the second degree, a variable parameter.

As an example, let us find the envelope of the straight

lines given by the equation

m^x â€” my + a = (1),

where m is a quantity which, by its variation, gives the

series of straight lines.

ENVELOPES. 409

If (1) pass through the fixed point (A, ^), we have

w^h â€” mk + a = (2).

This is an equation giving the values of m correspond-

ing to the straight lines of the series which pass through

the point (A, li). There can therefore be drawn two

straight lines from (A, ^) to touch the required envelope.

As (A, Aj) moves nearer and nearer to the required

envelope these two tangents approach more and more

nearly to coincidence, until, when (A, h) is taken on the

envelope, the two tangents coincide.

Conversely, if the two tangents given by (2) coincide,

the point (A, U) lies on the envelope.

Now the roots of (2) are equal if l? = 4:ah,

so that the locus of (A, k), i. e. the required envelope, is the

parabola y^ = 4:ax.

Hence, more simply, the envelope of the straight line (1 )

is the curve whose equation is obtained by writing down

the condition that the equation (1), considered as a quad-

ratic equation in m, may have equal roots.

By writing (1) in the form

a

y = mx H â€” ,

m

it is clear that it always touches the parabola y^ = ^ax.

In the next article we shall apply this method to the

general case.

434. To find the envelope of a straight line whose

equation involves^ in the second degree, a variable parameter.

The equation to the straight line is of the form

X^P-{-\Q + R = (1),

where X is a variable parameter and P, Q^ and M are

expressions of the first degree in x and y.

Equation (1) may be looked upon as an equation

giving the two values of X corresponding to any given

point T.

410 COORDINATE GEOMETRY.

Through this given point two straight lines to touch the

required envelope may therefore be drawn.

If the point T be taken on the required envelope, the

two tangents that can be drawn from it coalesce into the

one tangent at T to the envelope.

Conversely, if the two straight lines given by (1)

coincide, the resulting condition will give us the equation

to the envelope.

But the condition that (1) shall have equal roots is

Q^ = 4.PR (2).

This is therefore the equation to the required envelope.

Since P, Q, and R are all expressions of the first degree,

the equation (2) is, in general, an equation of the second

degree, and hence, in general, represents a conic section.

The envelope of any straight line, whose equation

contains an arbitrary parameter and square thereof, is

therefore always a conic.

435. The method of the previous article holds even if

P, Q, and R be not necessarily linear expressions. It

follows that the envelope of any family of curves, whose

equation contains a variable parameter X, in the second

degree, is found by writing down the condition that the

equation, considered as an equation in X, may have equal

roots.

436. Ex. 1. Find the envelope of the straight line which cuts off

from two given straight lines a triangle of constant area.

Let the given straight lines be taken as the axes of coordinates and

let them be inclined at an angle w.

The equation to a straight line cutting off intercepts / and g from

the axes is

rl=' <^>-

If the area of the triangle cut off be constant, we have

\f .g . sin w = const.,

i.e, fg=zconst.=K^ (2).

On substitution for g in (1), the equation to the straight line

becomes f'y -fIO + K^x=:0.

ENVELOPES. EXAMPLES. 411

By the last article, the envelope of this line, for different values of

/, is given by the equation

I.e. xy=-^.

The result is therefore a hyperbola whose asymptotes coincide with

the axes of coordinates.

Ex. 2. Find the envelope of the straight line ivhich is such that

the product of the perpendiculars draiun to it from two fixed points is

constant.

Take the middle point of the line joining the two fixed points as

the origin, the line joining them as the axis of x, and let the two

points be {d, 0) and {-d, 0).

Let the variable straight line have as equation

y=mx + c.

The condition then gives

md + c -md + c , , ^â€ž

X = constant = ^%

jjl + m^ Jl + m^'

so that c^ - viH^ =A^{1+ m^) .

The equation to the variable straight line is then

y-'mx=c= fJ{A'^ + d^)m^ + A^.

Or, on squaring,

mP (a;2 -A^- d^) - 2mxy + (y^ - A^) = 0.

By Art. 435, the envelope of this is

[2xyf = 4 (a;2 - ^2 _ ^2) (^2 _ ^2) ^

This is an ellipse whose axes are the axes of coordinates and whose

foci are the two given points.

Ex. 3. Find the envelope of chords of an ellipse the tangents at the

end of ivhich intersect at right angles.

Let the ellipse be â€” â€ž + ^r, = 1.

a^ 0^

If the tangents intersect at right angles, their point of intersection

P must lie on the director circle, and hence its coordinates must be of

the form (c cos ^, c sin 6), where c = Ja'^+ h^.

The chord is then the polar of P with respect to the ellipse, and

hence its equation is

x . c cos 6 V .c sin d ^

412 COORDINATE GEOMETRY.

a

Let t^tan - . Then since

l-*an2- ^_^2 2(

1 + tan^ -

the equation to the line is

ex 1-t'^ cy 2t _^

The envelope of this is (Art. 434),

^2g2 ^2g2

^2 2/^ _i

Since -:; â€” r-, s â€” r^ = a^ - &2 this equation represents a conic

a^ + b^ a^ + b^

confocal with the given one.

Ex. 4. The normals at four points of an ellipse meet in a point ;

if the line joining one pair of these points pass through a fixed point,

prove that the line joining the other pair envelopes a parabola which

touches the axes.

Let the equation to the ellipse be

%*t=^ W'

and let the equation to the two pairs of lines through the points be

lx+m7j = l (2),

and lj^x + miy = l (3).

By Art. 412, Cor. (1), we then have

11-^= â€” 2 ^^^ *^%=~i^ W*

If the straight line (3) pass through the fixed point (/, g), we have

BO that, by (4), -X_^|_=l,

and therefore 1= â€” ^ -r^ .

ENVELOPES. EXAMPLES. 413

If this value of I be substituted in (2), it becomes

m^a%^y + m {a?gy - W-fx - a^ft^) - d^g = 0,

the envelope of which is

(a^gy - hjx - a%Y = - ^a?g . a^&s^ ,

i.e, {a^gy - V^fxf + 2.a%^ {bjx + a?gy) + a%'^=0 (5) .

This is a parabola since the terms of the second degree form a

perfect square. Also, putting in succession x and y equal to zero, we

get perfect squares, so that the parabola touches both axes.

437. To find the envelope of the straight line

lx + TYiy + n â€” ^ (1),

where the quantities I, m, and n are connected hy the

relation

aP + hm? + cn^ + Ifmn + "Ignl + Vilm = (2).

[Equation (1) contains two independent parameters â€”

and - , whilst (2) is an equation connecting them. We

n

could therefore solve (2) to give - in terms of â€” : on sub-

^ ' n n

stituting in (1) we should then have an equation containing

one independent parameter and its envelope could then be

found.

It is easier, however, to proceed as follows.]

Eliminating n between (1) and (2), we see that the

equation to the straight line may be written in the form

aV' + htn^ â– \-g(}x-\- myY â€” 2 {fm + gl) (Ix + my) + 2hlm = 0,

/ ^\^ I

i. e. (a â€” 2gx + cx^) ( â€” j + 2 {cxy â€” gy -fx + h) â€”

^{h-2fy + cf) = 0.

The envelope of this is, by Art. 435,

{cxy - gy -fx + hf = {a- 2gx + ex") (b - 2fy + cy\

i.e., on reduction,

x^ (be -D + y' (ca - g') + 2xy (fg - ch)

+ 2x{fh-bg)^-2y{gh-af) + ab-h'' = 0.

414 COORDINATE GEOMETRY.

The envelope is therefore a conic section.

Cor. The envelope is a parabola if

i. e. if c = 0, or if abc + %fgh â€” af^ â€” hg^ - ch^ ~ 0.

438. Ex. Find the envelope of all chords of the parabola y^ = 4aa;

lohich subtend a given angle a at the vertex.

Any straight line is

Za3 + m7/ + w = (1).

The lines joining the origin to its intersections with the parabola

are, (by Art. 122), ny^ = - 4:ax {Ix + my) ,

i.e. ny^ + 4:a'mxy + 4alx^=0.

If a be the angle between these lines, we have

2/J'ia^m^-4aln

tana=â€” ^^ j-z ,

n+4:al

i.e. 16a2Z2 - IQa^ cot^ am^+n^ + 8aln{l + 2 cot^ a) = 0.

With this condition the envelope of (1) is, by the last article,

a;2 ( - 16a2 cot2 a) + 2/2 [I6a2 _ (4^ + 8a cot2 a)2]

+ 2x . 16a2 cot2 a (4a + 8a Cot2 a) - 256a4 cot2 a= 0,

i.e. the ellipse

[a; - 4a (1 + 2 cot2 a)]2 + 4 cosec2 a . 2/2= 64 cot2 a . cosec2 a.

EXAMPLES. XLIX.

Find the envelope of the straight line - + ^=1 when

a p

1. aa + b^=c. 2. a + ^+s/aF+^=c.

b^ a^

^' ^2 + ^2=1-

Find the envelope of a straight hne which moves so that

4. the sum of the intercepts made by it on two given straight

lines is constant.

5. the sum of the squares of the perpendiculars drawn to it from

two given points is constant.

6. the difference of these squares is constant.

7. Find the envelope of the straight line whose equation is

ax cos 9 + by sin 6 = c^.

[EXS. XLIX.] ENVELOPES. EXAMPLES. 415

8. Circles are described touching each of two given straight lines ;

prove that the polars of a given point with respect to these circles all

touch a parabola.

9. From any point P on a parabola perpendiculars P3I and FN

are drawn to the axis and tangent at the vertex; prove that the

envelope of MN is another parabola.

10. Shew that the envelope of the chord which is common to the

parabola y^=:4:ax and its circle of curvature is the parabola

y^ + 12ax = 0.

11. Perpendiculars are drawn to the tangents to the parabola

y^=4:ax at the points where they meet the straight line x = b; prove

that they envelope another parabola having the same focus.

12. A variable tangent to a given parabola cuts a fixed tangent in

the point A ; prove that the envelope of the straight line through A

perpendicular to the variable tangent is another parabola.

13. Shew that the envelope of chords of a parabola the tangents

at the ends of which meet at a constant angle is, in general an ellipse.

14. A given parabola slides between two axes at right angles ;

prove that the envelope of its latus rectum is a fixed circle.

15. Prove that the envelope of chords of an ellipse which subtend

a right angle at its centre is a concentric circle.

16. If the lines joining any point P on an ellipse to the foci meet

the curve again in Q and R, prove that the envelope of the line QE is

the concentric and coaxal ellipse

x^ y^ fl + ey_

17. Prove that the envelope of chords of the rectangular hyperbola

xy=a^, which subtend a constant angle a at the point (a?', y') on the

curve, is the hyperbola

x^x'^ + yh^'^=2a^xy (1 + 2 cot^ a) - 4a^ cosec^ a.

18. Chords of a conic are drawn subtending a right angle at a

fixed point 0. Prove that their envelope is a conic whose focus is

and whose directrix is the polar of with respect to the original conic.

19. Shew that the envelope of the polars of a fixed point with

respect to a system of confocal conies, whose centre is 0, is a parabola

having GO as directrix.

20. A given straight line meets one of a system of confocal conies

in P and Q, and BS is the line joining the feet of the other two

normals drawn from the point of intersection of the normals at P and

Q ; prove that the envelope of RS is a parabola touching the axes.

416 COORDINATE GEOMETRY. [ExS. XLIX.]

21. ABGD is a rectangular sheet of paper, and it is folded over so

that G lies on the side AB ; prove that the envelope of the crease so

formed is a parabola, whose focus is the initial position of G.

22. A circle, whose centre is A, is traced on a sheet of paper and

any point B is taken on the paper. If the paper be folded so that the

circumference of the circle passes through B, prove that the envelope

of the crease so formed is a conic whose foci are A and B.

23. In the conic -=l-ecos^ find the envelope of chords which

r

subtend a constant angle 2a at the focus.

24. Circles are described on chords of the parabola y^ = 'iax, which

are parallel to the straight line lx + my = 0, as diameters; prove that

they envelope the parabola

[ly + 2maf = 4a (Z^ + m^) {x + a).

25. Prove that the envelope of the polar of any point on the circle

{x-\-af+{y + hf=h'^ with respect to the circle x^+y^=c^ is the conic

h^ (a;2 + 2/2) = [ax + hy + c^-

26. Chords of the conic -=l-ecos^ are drawn passing through

r

the origin and on these circles as diameters circles are described.

Shew that the envelope of these circles is the two circles

- â€” h ecoBd ) = lÂ±e.

r \r J

ANSWERS.,

I. (Pages 14, 15.)

1. 5. 2. 13. 3. 3V7. 4. s/aJ+b\

5. ^a2 + 2&2 + c2-2afc-26c. 6. 2a sin -^^.

7. a {j)x^ - m^) sj{m^ + m.^f + 4. 9. 3Â±2^15.

15. (Â¥->Â¥)â€¢ 16. (-2,-9). 17. (1, -I); (-11, 16).

18. (-5H, 2A); (-20i, 34|). 19. (-i,0); (-1, 2).

20. (-1,1); (1.1); (I, -I).

^^- \'a + b' ^+& 7' VÂ«-^ ' a-bj'

^^- \^ /c + Z + m ' h + l + m J'

II.' (Pages 18, 19.)

1. 10. 2. 1. 3. 29. 4. 2ac.

5. a?. 6. 2fl& sin ^IZ,^? gin ^^I^^ sin ^LlL^a .

i Jj a

7. a2(w2-Â»n3)(m3-mi)(mi-W2).

8. la^im^-m^im^-m-^ivi-^-m^.

9 . la'^ (mg - W3) (wig - wij) (m^ - mg) -f- mjin^m^ .

13. 201. 14, 96.

III. (Pages 22, 23.)

12. 2^5. 13. x/79. 14. s/7a. 16. 1(8-3^3).

17. '^-. 18. iÂ«V3- 25. r^=ci\ 26. ^=a.

27. r=2acose. 28. r cos 2^ = 2a sin ^. 29. r cos ^ = 2a sin^ ^.

30. r2 = a2 cos 2^. 31. x^ + 2f = a^. 32. y = mx.

33. a;2 + 2/2 = aa;. 34. {x^ + 2ff = 4:a^xY-

35. (a;2 + 2/2)2z=a2(a;2-2/2). 35, a;i/ = a2. 37. x'^-if = a\

38. 2/2 + 4flx = 4a2. 39. 2/2 = 4aa; + 4a2.

40. ^^ - 3x^2 ^ 3a;22^ -y^= 5kxy.

L. 27

11 COORDINATE GEOMETRY.

IV. (Page 30.)

8. 2ax + k^-=^0. 9. Oi2-l)(a;2 + ?/+a2) + 2aa;{w2 + l) = 0.

10. 4a;2(c2-4a2) + 4cV = c2(c2-4a2). H. {Qa-2c) x = a^-c\

12. ^f-i^J-2x + 5 = 0. 13. ^y + 2x + S = 0. 14. x + y = l.

15. y = x. 16. ?/ = 3.r. 17. 15a;2-2/2 + 2aa; = a2.

18. .r2 + 2/2 = 3. 19. fl^2 + ^2^4y^

20. 8a:2 + 8?/2 + 6;r-36;c + 27=:0. 21. .'c2 = 3?/2.

22. X'-\-2ay = a'^.

23. (1) 4a;2+3?/2 + 2a2/ = a2; (2) x'' -^if + Say = 4.a^.

V. (Pages 41, 42.)

1. y=x + l. 2. a;-2/-5 = 0. 3. x-y >^^-2^2> = 0.

4. 52/-3a; + 9=:0. 5. 2a; + 3y = 6. 6. 6a; - 5?/ + 30 = 0.

7. (1) a; + 2/ = ll; (2) 2/-a;=l. 8. x^y + 1 = ; x-y = ^.

9. a;^' + a:V = 2.-cy. 10. 20?/ -9a; = 96. 15. x-\-y=0.

16. 2/-^ = l- 17. ly + 10x = ll.

18. ax - by = (lb. 19. (Â«-2&) a;- &?/ + &^ + 2a&- a2 = 0.

20. ^j(Â«i + i2)-2a;=:2fl^if2. 21. Â«ii22/ + ^ = Â« {*i + ^2)-

22. X cos 1 (01 + 02) + 2/ sin i (01 + 02) = a cps J (01 - 02).

Â«Â« ^ 01 + 02 2/ â€¢ 01 + 02 01 ~ 02

23. -cos?^+|sin^^=cos'^-^ - ^.

24. &a; cos 4 (01 - 02) - Â«2/ sin J (0i + 02) = ab cos i (0i + 02).

25. x + Sy + 7 = 0; y-3x=l; y + 7x = ll.

26. 2x-Sy = 4:; y-Sx=l; x + 2y = 2.

27. 1/ (a' -a)-x {V -b) = a'b - ah' ; y {a' - a) J^x{b'-b) = a'b' - ab.

28. 2ay-2b'x = ab-a'b'. 29. y = ^x\ 2?/ = 3x.

VI. (Pages 48, 49.)

1. 90Â°. 2. tan-iff. 3. tan-i-f. 4. 60Â°.

8. 4y + 3a; = 18. 9. 7?/-8rc = 118. 10. 4i/ + lla;=10.

11. a; + 4?/ + 16 = 0. 12. a.r + ftr/ = a\

13. 2a; (a - a') + 2?/ (6 - &') = a^ - a"^ + 62- y^.

15. yx' -xy' = 0; a^xy' -b^x'y = {a^-b^)x'y' ; xx' -yy' = x' - y'^.

16. 121y-88a; = 371; 33?/ -24a; =1043.

17. a; = 3; 2/ = 4; 4i. 19. a; = ; 2/ + x/3a; = 0.

20. 2/ = ^; (l-m2)(?/-7f) = 2m(a;-/i).

21. tan-iH; 9a;-7i/ = l; 7a; + 9?/ = 73.

ANSWEES. m

VII. (Pages 53, 54.)

1. ^. 2. 2f. 3. 5^. 4.

5. a cos J (a -/S). 8.

a^ + ab-b^

Ja^ + b'-^

^1 + m^

9. |^(&Â±V^MT^), ol. 11. 4(2 + ^3).

VIII. (Pages 61-65.)

/ - 11 41\ _a&_ a& \

â– 'â– â€¢ V 29" '297* a + 6 'Â« + &;â€¢

,4. {a cos i (01 + 02) sec i (01 -0o), asini (01 + 0^)860 i(0i- 0^)}.

/ a{b-b') 2bb' \ 130

^" \ b + b' ' b + b'J' ^' 17^29'

8. y = a', %y = 4.x + ^a. 9. (1,1); 45Â°.

10. (f,i); tan-160. 11. (-1, -3); (3,1); (5,3).

12. (2, 1); tan-i,^. 13. 45Â°; (-5, 3) ; a;- 3^ = 9; 2.r-?/ = 8.

14. 3 and - f . 19. w?! (Â«2 ~ ^3) + ^% (^3 ~ '^1) + "^3 (^1 ~ ^2) = 0-

20. (-4,-3). 21. (h/-!^)- ^ 23. 43.i;-29y = 71.

24. x-y=ii. 25. 2/ = 3jc. 26. y = x.

27. a^y-b^x=ab{a-b). 28. 3;r + 42/ = 5a. 29. a; + ?/ + 2 = 0.

30. 23a; + 23^ = 11. 31. 13.r - 23?/ = 64.

33. ^ic + 5j/ + + X (^'a; + J5 '?/ + C) = where A is

(1) -^,, (2) -;g7, (3) -^, - ^, and (4) - ^.^.^^y^ ^..

37. y = 2', x = 6. 38. 99a; + 77?/ + 71 = 0; 7a;-9y-37=0.

39. x-2y + l = 0; 2x + y=:S.

40. ^(2V2-3)+?/(x/2-l) = 4V2-5;

a;(2V2 + 3) + ?/(V2 + l) = 4^2 + 5.

41. {y-b){m + m') + {x-a){l-mm') = 0;

[y - 6) (1 - ?u7?i') -{x- a) (m + ?/t') = 0.

42. 33a; + 9?/ = 31; 112ic- 64?/ + 141 = 0; 7y-x = 18.

43. a;(3+V17)+2/(5 + V17) = 15 + 4^17;

a;(4 + V10) + ?/(2 + ^10) = 4VlO + 12;

a; (2 V34 - 3 V5) +2/ ( V34 - 5V5) = 6 V34 - 5 V5.

44. A{y-k)-B{x-h)=^{Ax + By + C).

45. At an angle of 15Â° or 75Â° to the axis of x.

27â€”2

IV COORDINATE GEOMETRY.

IX. (Pages 72, 73.)

I. (1) tan-i^; (2) 15Â°. 2. tan"!^^.

3. tan-i^^^tancj.

7, y = 0, y = x-a, x = 2a, y = 2a, y = x + a, x = 0, y = x, x = a, and

y = a, where a is the length of a side.

10. 2/(6-V3) + ^(3V3-2) = 22-9V3. 11. h

12. 10?^ - 11a; + 1 = ; ^^ ^IH-

X. (Pages 78â€”80.)

4. (-7,3). 5. (-ii, If); m-

/-85 + 7V5 7V5-27 \ 35-7^5 â€ž n^.^.^.

^- \ 120"""' W^'J' 120" â€¢ 'â€¢ ^^'S'^*'"^?-

(6 + ^/10 2 + VlO ] / 6-^10 2-V10\ /8-VlO lo + ^lO N

Â°- I 2 ' 2 j'V 2 ' 2"y'V 6 ' 6 J'

9. {%, f), (2, 12), (12, 2), and ( - 3, - 3) ; 1^2, 4^2, 4^2, and 6^2.

10. (-13^,194). 11. 4. 12. 7||. 13. f.

14. ^. 15. M&-c)(c-a)(a-&).

16. Â«^ (w^a - ^'^3) {>^h - ''^h) (^'h " 'm2)^2m^m^m^.

,Â« lf(C2-Cs)^ (Cs-Ci)2 (Ci-Co)2]

17. i(Ci-Co)2-f K-wi2- 18. oV-^ ^ + i-3 iL + U 2^1

23. (f,f)-

24. 10i/ + 32Â« + 43 = 0; 25a; + 29?/ + 5 = 0; 2/ = 5a; + 2; 52a; + 80i/ = 47.

26. (4 + iv/3, f + v/3) ; (4 + i^/3, f + W3).

XI. (Pages 85â€”87.)

1. a;2 + 2a;?/cota-2/2 = a2, 2. ?/2 + Xa;2=:Xa2.

3. (7?i + l)a;=(m-l)a. 4. (w + ?i) (a;2 + ?/'-' + a^)- 2aa;(m-/Â«) = c2.

5. .T + 2/ = csec2|. 6. a;-?/ = <icosec2-.

7, j; + ?/ = 2ccosecw. 8. 2/-^ = 2ccosecw,

9. a:2 + 2a;i/cosw + 2/^=4c2cosec2w.

10. (^^ + y^) cos w + a;?/ (1 + cos^ w) = a; [a cos w + &) + 2/ (& cos w + a).

II. a;(m + cosw)+i/(l+?Â«cosw) = 0.

12. (i) ai/2 + &a;2+(a + &)ic?/-a?/(a + 2&)-&a;(2rt + 6) + a&(a + 6) = 0;

(ii) y=a;. 19. A straight line.

20. A circle, centre 0. 25. A straight line.

27. If P he the point [h, k), the equation to the locus of S is

h k ,

- + - = 1.

a; 2/

ANSWERS. V

XII. (Page 94.)

1. {x-Sy){x-'iy) = 0;ta.n-'^^\. 2. {2.t- ll?/)(2a;-?/) = 0; tan-i -J.

3. (lla; + 2y)(3a;-7i/)=0; tan-iff. 4. x = l ; x = 2', x = 3.

5. 2/= Â±4. 6. {y + ^x){y-2x){y-Sx) = 0;ia.n-^{-^);t&n-^{}).

7. a;(l-sin^)+?/cos^ = 0; a; {l + sin^) + ^cos^ = 0; 0.

8. r/siii^ + a;cos^= Â±a;^/cos2^; tan-^ (cosec^ Aycos2^).

9. 12x^-7xy-12y^ = 0; Ux^ + Hxy -71y^ = 0\ x^-y"^0',

x^-y^=0.

XIII. (Pages 98, 99.)

1, (I, -Â¥);45Â«. 2. (2, 1); tan-if. 3. (-f, -t);90Â°.

4. (-1, 1); tan-13. 6. -15. 7. 2. 8. -10 or -171.

9. -12. 10. 6. 11. 6. 12. 14. 13. -3.

14. fori=f. 16. (i) c{a + h) = 0; {ii) e = 0, ox ae = bd.

17. Siz + e.'c^se; 5?/-6a;=14.

XV. (Page 112.)

1. (1) i/'2=4a;'; (2) 2x'^ + y"' = 6.

2. (1) a;'2 + 2/'2=:2ca;'; (2) x'^ + y'^=2cy\

3. (a-&)2(a;'2 + ^'2) = a262.

4. (1) 2x'y' + a^=0; 9x'^ + 25y'^ = 225', x'^ + y'^ = l.

5. x"^ + 2j"^ = r^', x'^-y'^=a^cos2a. 6. a;'^ - 4?/'2 = ^2.

8. tan-ij; -C-^^^M^a,

XVI. (Page 117.)

1. 2x'-^Qy' + l=::0. 2. a;'2 + ^3a;y = l. 3. x'^ + y'^-=8.

4. ?/'2=4a;'cosec2a.

XVII. (Pages 123â€”125.)

I x^ + y^ + 2x-4:y = L 2. a;2 + ?/2+10a; + 12?/ = 39.

3^ x^ + y^-2ax + 2hy = 2ab. 4. a;^ + 7/2 + 2fla; + 26i/ + 262 = 0.

5. (2,4); ^61. 6. (f, l);ix/l3. 7. (^ o) 5 ^ ^-

8. {o,-f);Jp+g'- 9. (-7 - ^. -7tT=0 ' '"

13. 15a;2 + 15?/2- 94.^ + 18^ + 55=0.

14. &(a;2 + 2/2-a2) = a;(&2 + ;i2_^2), ^5^ a:2 + 7/ - aa; - fcy = 0.

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