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S. L. (Sidney Luxton) Loney.

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(by Art. 412, Cor. 2)

mim2=-g (3),

1^

B

If (/, g) be the fixed point through which PQ passes, we have

g=m^f+Cj^ (5).

Now the middle point of BS lies on the diameter conjugate to it,
i.e. by Art. 376, on the diameter

A



and c-^c^=. - — (4).



Bm,



i.e., by (3), y - m^x (6).



890 COORDINATE GEOMETRY.

Now, from (4) and (5),



c„= -



B{g-M)'
so that, by (3), the equation to BS is

y=mr,''-B{g-fm,) - ^'^^-

Eliminating m^ between (6) and (7), we easily have, as the equation
to the required locus,

{Ax^ + By^){gx+fy)+xy = 0.

Cor. From equation (6) it follows that the diameter conjugate to
ES is equally inclined with PQ to the axis, and hence that the points
P and Q and the ends of the diameter conjugate to BS are coney clic
(Art. 400),

EXAMPLES. XLVI.

1. If the sum of the squares of the four normals drawn from a
point to an ellipse be constant, prove that the locus of is a conic.

2. If the sum of the reciprocals of the distances from a focus of
the feet of the four normals drawn from a point to an ellipse be

4
^-r r > prove that the locus of is a parabola passing through that

focus.

3. If four normals be drawn from a point to an ellipse and if
the sum of the squares of the reciprocals of perpendiculars from the
centre upon the tangents drawn at their feet be constant, prove that
the locus of is a hyperbola.

4. The normals at four points of an ellipse are concurrent and
they meet the major axis in Gj, G^, G^a, and G^; prove that

C(?i "*" CG^ "^ GG^'^CG^~ GG^+GG^+GG^+GG^'

5. If the normals to a central conic at four points L, M, N, and
P be concurrent, and if the circle through L, 31, and N meet the curve
again in P', prove that PP' is a diameter.

6. Shew that the locus of the foci of the rectangular hyperbolas
which pass through the four points in which the normals drawn from
any point on a given straight line meet an ellipse is a pair of conies.

7. If the normals at points of an ellipse, whose eccentric angles
are a, j8, and y, meet in a point, prove that

sin (j8 + 7) + sin (7 + a) + sin (a + iS)=0.
Hence, by page 235, Ex. 15, shew that if PQB be a maximum
triangle inscribed in an ellipse, the normals at P, Q, and B are
concurrent.



[EXS. XLVI.] CONCURRENT NORMALS. EXAMPLES. 391

8. Prove that the normals at the points where the straight line

X y x^ v^

+ i~?- — = 1 meets the ellipse -^ + ^ =1 meet at the point

a cos a &sma ^ a^ b^



I - ae^ cos-* a, -J— sin** a 1



9. Prove that the loci of the point of intersection of normals at
the ends of focal chords of an ellipse are the two ellipses

aY ( 1 + e2)2 + 62 (a; ± ae) {x t ae^) = 0.

10. Tangents to the ellipse —2 + j^ = ^ are drawn from any point

on the ellipse -2 + f^=^; prove that the normals at the points of
contact meet on the ellipse a^x^ + bY=^{a^-b^)^.

11. . Any tangent to the rectangular hyperbola 4:xy=ab meets the
ellipse -g + |j=l in the points P and Q; prove that the normals at P
and Q meet on a fixed diameter.

12. Chords of an ellipse meet the major axis in the point whose

distance from the centre is a \ / ; prove that the normals at its

V a + b

ends meet on a circle.

13. From any point on the normal to the ellipse at the point
whose eccentric angle is a two other normals are drawn to it ; prove
that the locus of the point of intersection of the corresponding
tangents is the curve

xy + bxsina + ay cosa = 0.

14. Shew that the locus of the intersection of two perpendicular
normals to an ellipse is the curve

(a2 + 62) (a;2 + ^2) (^2^2 + 62^.2)2 _ (^2 _ ^2)2 (^2^2 _ 62^.2)2^

/i*2 nj2

15. ABC is a triangle inscribed in the elUpse —2 + 72=1 having

each side parallel to the tangent at the opposite angular point ; prove
that the normals at A, B, and G meet at a point which lies on the
eUipse a2a;2 + 62^2 = ^ (a2 - 62)2.

16. The normals at four points of an ellipse meet in a point {h, k).
Find the equations of the axes of the two parabolas which pass
through the four points. Prove that the angle between them is

2 tan-i - and that they are parallel to one or other of the equi-con-

jugates of the ellipse.



892 COORDINATE GEOMETRY. [EXS. XLVI.]



17. Prove that the centre of mean position of the four points on
le ellipse -o + |o =
(a, /3), is the point



the ellipse -2 + I2 — ^» *^® normals at which pass through the point






18. Prove that the product of the three normals drawn from any
point to a parabola, divided by the product of the two tangents from
the same point, is equal to one quarter of the latus rectum.

19. Prove that the conic 2aky = {2a-h)y^ + 4:ax^ intersects the
parabola y^=4:ax at the feet of the normals drawn to it from the point
{h, k).

20. From a point {h, k) four normals are drawn to the rectangular
hyperbola xy = c^; prove that the centre of mean position of their feet

(h k\
- , 2 ) J aiid that the four feet are such that each is the

orthocentre of the triangle formed by the other three.

Confocal Conies.

415. Def. Two conies are said to be confocal when
they have both foci common.

To find the equation to conies which are confocal with

the elli2^se

2 2

All conies having the same foci have the same centre
and axes.

The equation to any conic having the same centre and
axes as the given conic is

?4=i ••••■••;;;;;; (^)-

The foci of (1) are at the points {^\Ja^ — h^^ 0).
The foci of (2) are at the points {^sJA - B^ 0).
These foci are the same if

A-B = a^-h%

i.e.ii A-a' = B-b''^X (say).

.*. A^a^ + \, and B = ¥ + X.



CONFOCAL CONICS. 393



The equation (2) then becomes

^ 19 . \ -^5



which is therefore the required equation, the quantity A,
determining the particular confocal.

416. For different values of X to trace the conic given
hy the equation

^ + -^-=1 (1).

First, let X be very great ; then a^ + X and 6^ + X are
both very great and, the greater that X is, the more nearly
do these quantities approach to equality. A circle of
infinitely great radius is therefore a confocal of the
system.

Let X gradually decrease from infinity to zero ; the
semi-major axis \J a'^ + X gradually decreases from infinity
to a, and the semi-minor axis from infinity to h. When X
is positive, the equation (1) therefore represents an ellipse
gradually decreasing in size from an infinite circle to
the standard ellipse

a" b^

This latter ellipse is marked / in the figure.

Next, let X gradually decrease from to — b^. The
semi-major axis decreases from a to \/a^ — b^, and the semi-
minor axis from b to 0.

For these values of X the confocal is still an ellipse,
which always lies within the ellipse /; it gradually
decreases in size until, when X is a quantity very slightly
greater than — b^, it is an extremely narrow ellipse very
nearly coinciding with the line SH, which joins the two
foci of all curves of the system.

Next, let X be less than — b^ ; the semi-minor axis
\/b^ + X now becomes imaginary and the curve is a hyper-
bola ; when X is very slightly less than — b^ the curve is a



394



COOKDINATE GEOMETRY.



hyperbola very nearly coinciding with the straight lines
SX and SX'.




[As X passes through the value — h^ it will be noted that
the confocal instantaneously changes from the line-ellipse
SH to the line-hyperbola SX and HX'.^

As X gets less and less, the semi-transverse axis Ja^ + \
becomes less and less, so that the ends of the transverse
axis of the hyperbola gradually approach to C, and the
hyperbola widens out as in the figure.

When X = — a^, the transverse axis of the hyperbola
vanishes, and the hyperbola degenerates into the infinite
double line TOY'.

When X is less than — a% both semi-axes of the conic
become imaginary, and therefore the confocal becomes
wholly imaginary.

417. Through any point in the plane of a given conic
there can he drawn two conies confocal with it; also one of
these is an ellipse and the other a hyperbola.

Let the equation to the given conic be

t + t^l
and let the given point be (f g).



CONFOCAL CONICS. 395

Any conic confocal with the given conic is



If this go through the point (/, g), we have

•^ +^^ = 1 (2).



This is a quadratic equation to determine X and there-
fore gives two values of X.

Put b^ + X = fi, and hence

a^ + X = fi + a^-b^ = fjL + aV.
The equation (2) then becomes

i.e. />c2 + /x(aV-/2-/)-^2^iV = (3).

On applying the criterion of Art. 1 we at once see that
the roots of this equation are both real.

Also, since its last term is negative, the product of
these roots is negative, and therefore one value of ju, is
positive and the other is negative.

The two values of b^ + X are therefore one positive and
the other negative. Similarly, the two values of a^ + X can
be shewn to be both positive.

On substituting in (2) we thus obtain an ellipse and a
hyperbola.

418. Confocal conies cut at right angles.
Let the confocals be

+ 70 . V ^ 1, and -y— T-+TT-r^=l'



«2 + Xi b'^ + X^ ' a'^ + X^ b'^ + X,

and let them meet at the point ix\ y').

The equations to the tangents at this point are
xx' yy' - , £C£c' yy' ^



396 COORDINATE GEOMETRY.

These cut at right angles if (Art 69)



+ 7JJ-^v€T^^v^ = ^ «•



But, since (x\ 2/) is a common point of the two confocals,
we have

^ + ^ - 1 and 4- ^ - 1

«2 + ^^ + 52 + x,~'' ^'^"^ a' + X, h' + X,~ '

By subtraction, we have

/2/_i l_Vv'^^-^ ^-A=o



a;-

/2



**^' (a^ + X^ {a\-\- X,) ■*" (F+ Aj) (6^ + X,) " ^ ^*

The condition (1) is therefore satisfied and hence the
two confocals cut at right angles.

Cor. From equation (2) it is clear that the quantities
b^ + A-i and b^ + X^ have opposite signs ; for otherwise we
should, have the sum of two positive quantities equal to
zero. Two confocals, therefore, which intersect, are one an
ellipse and the other a hyperbola.

419. One conic and only one conic, confocal with the conic
-2 + 12 = I5 ^^^ ^^ drawn to touch a given straight line.

Let the equation to the given straight line be

X cos a + y sin a =p (1) .

Any confocal of the system is



x^ , y



+ ./:^ = l (2).



The straight line (1) touches (2) if

2j2= {a? + X) cos2 a + (&2 + X) sin2 a (Art. 264) ,
i.e. if X =_p2 _ a? cos^ a - 6^ sin2 ^^

This only gives one value for X and therefore there is only one
conic of the form (2) which touches the straight line (1).

Also X + a,2=^2^^^2_ j2^ gin2 jj_a j.gal quantity. The conic is
therefore real.



CONFOCAL CONICS. EXAMPLES. 397



EXAMPLES. XLVII

1. Prove that the difference of the squares of the perpendiculars
drawn from the centre upon parallel tangents to two given confocal
conies is constant.

2. Prove that the equation to the hyperbola drawn through the
point of the ellipse, whose eccentric angle is a, and which is confocal
with the elhpse, is



cos^ a sin^ a

3. Prove that the locus of the points lying on a system of confocal
eUipses, which have the same eccentric angle a, is a confocal hyperbola
whose asymptotes are inclined at an angle 2a.

4. Shew that the locus of the point of contact of tangents drawn
from a given point to a system of confocal conies is a cubic curve,
which passes through the given point and the foci.

If the given point be on the major axis, prove that the cubic
reduces to a circle.

5. Prove that the locus of the feet of the normals drawn from a
fixed point to a series of confocals is a cubic curve which passes
through the given point and the foci of the confocals.

6. A point P is taken on the conic whose equation is



such that the normal at it passes through a fixed point {h, k); prove
that P lies on the curve

1 — ? — = .

y -k x-h hy-kx

7. Two tangents at right angles to one another are drawn from
a point P, one to each of two confocal ellipses ; prove that P Hes on
a fixed circle. Shew also that the line joining the points of contact is
bisected by the line joining P to the common centre.

8. From a given point a pair of tangents is drawn to each of a
given system of confocals ; prove that the normals at the points of
contact meet on a straight Une.

9. Tangents are drawn to the parabola y^=4iXs/oP-h^, and on
each is taken the point at which it touches one of the confocals



a^ + \ b^+\
prove that the locus of such points is a straight line.



398 COORDINATE GEOMETRY. [Exs. XLVII.]

10. Normals are drawn from a given point to each of a system of
eonfocal conies, and tangents at the feet of these normals ; prove that
the locus of the middle points of the portions of these tangents
intercepted between the axes of the confocals is a straight line.

11. Prove that the locus of the pole of a given straight line with
respect to a series of confocals is a straight line which is the normal
to that eonfocal which the straight line touches.

12. A series of parallel tangents is drawn to a system of eonfocal
conies ; prove that the locus of the points of contact is a rectangular
hyperbola.

Shew also that the locus of the vertices of these rectangular
hyperbolas, for different directions of the tangents, is the curve
r'^ = c^eos2d, where 2c is the distance between the foci of the
confocals.

13. The locus of the pole of any tangent to a eonfocal with respect
to any circle, whose centre is one of the foci, is obtained and found to
be a circle ; prove that, if the circle corresponding to each eonfocal be
taken, they are all coaxal.

14. Prove that the two conies

ax^ + 2hxy + by^=l and a'x^ + 2h'xy + b'y^=l
can be placed so as to be eonfocal, if

(ah-h?f ~ {a'b'-h'Y '

Curvature.

420. Circle of Curvature. Def. If F, Q, and R

be any three points on a conic section, one circle and only
one circle can be drawn to pass through them. Also this
circle is completely determined by the three points.

Let now the points Q and R move up to, and ultimately
coincide with, the point P ; then the limiting position of
the above circle is called the circle of curvature at P ; also
the radius of this circle is called the radius of curvature at
Pj and its centre is called the centre of curvature at P.

421, Since the circle of curvature at P meets the
conic in three coincident points at P, it will cut the curve
in one other point P'. The line PP' which is the line
joining P to the other point of intersection of the conic and
the circle of curvature is called the common chord of
curvature.



I



CIRCLES OF CURVATURE. 399

We shewed, in Art. 400, that, if a circle and a conic
intersect in four points, the line joining one pair of points
of intersection and the line joining the other pair are
equally inclined to the axis. In our case, one pair of
points is two of the coincident points at P, and the line
joining them therefore the tangent at P ; the other pair of
points is the third point at P and the point P', and the
line joining them the chord of curvature PP'. Hence the
tangent at P and the chord of curvature PP' are, in any
conic, equally inclined to the axis.

4t^2i, To find the equation to the circle of curvature and
the length of the radius of curvature at any point (af, 2at)
of the parabola y"^ = 4:ax.

If S=0 be the equation to a conic, T=0 the equation
to the tangent at the point P, whose coordinates are at^ and
2at, and L = the equation to any straight line passing
through P, we know, by Art. 384, that jS + \.L.T=0 is
the equation to the conic section passing through three
coincident points at P and through the other point in which
X = meets aS^-0.

If A. and L be so chosen that this conic is a circle, it will
be the circle of curvature at P, and, by the last article, we
know that L = will be equally inclined to the axis with
T=0.

In the case of a parabola

S = 2/2 - iax, and T=ty-x- af. (Art. 229.)

Also the equation to a line through {af, 2at) equally
inclined with :Z' = to the axis is

t(y— 2at) + x — at'~0,

so that L ^ ty + X — Sat^.

The equation to the circle of curvature is therefore

y^ — 4ax ■{■ \(ty — X — af) {ty + x — ^af) = 0,

where 1 + Xi^ = — A., i.e. \ = —^ ^ •

I +f



400 COORDINATE GEOMETRY.

On substituting this value of A,, we have, as the required
equation,

x^ + y'- "lax {W + 2) + iayt^ - ZaH"" = 0,

i.e. [x-a{2 + 3f)J + [y + 2at^f = ia? (1 + ff.

The circle of curvature has therefore its centre at
the point (2a + ^af, — 2af) and its radius equal to

2a (1 + ff.

Cor. If S be the focus, we have SP equal to a + at\ so

2 . SP^
that the radius of curvature is equal to



V



a



423. To find the equation to the circle of curvature at

the 'point P (a cos <^, h sin ^) of the ellipse -3 + rg = 1-

a

The tangent at the point P is

X It

— cos <f> + T sin d) = 1.
a

The straight line passing through P and equally inclined
with this line to the axis is

cos<^. ,. sin<^ I ' ,\ n

(x — a cos ch) — ^ (y — o sm d>) = 0,

a

X 1/

i. e. - cos </> — r sin d> — cos 2d) = 0.

a

The equation to the circle of curvature is therefore of
the form

,2



^ y^



1 + X - cos ^ + f sin ^ — 1
- cos </) — ^ sin (^ - cos 2</) =0 (1).



Since it is a circle, the coefficients of x'^ and y^ must be
equal, so that

1 cos^ ^ 1 . sin^ <^

-g + A r— = 77; — A

and therefore X =



6^ cos^ cf> + a^ sin^ ^ *



CIRCLES OF CURVATURE. 401

On substitution in (1), the equation to the circle of
curvature is

+ («^ - b') [^ cos^ .f,-^ sin^ <l> - ^^ (1 + cos 2,^)

+ z (1 - cos 2cf>) + cos 2(j>\ = 0,

+ ^2 (cos^ <^ - 2 sin^ <^) - &2 (2 cos^ </, - sin^ </>) - 0.
The equation to the circle of curvature is then

^x -— cos^ ^1 + h/+ ~y- sm^ ^1

+ &^{2cos2<^-sin2^}

(a^ sin^ (i) + 6^ cos^ 6Y i, , .

= ^7^ , alter some reduction.

The centre is therefore the point whose coordinates are

cos^ ^, 7 — sin^ ^ j and whose radius is

(g^ sin^ <j> + ¥ cos^ <j!>) ^

Cor. 1. If Ci) be the semi-diameter which is conju-
gate to CP, then D is the point (90° + <^), so that its
coordinates are — a sin ^ and b cos <^. (Art. 285.)

Hence CD^ = a^ sin^ <^ + ¥ cos^ ^,

and therefore the radius of curvature p = — 5— .

ao

Cor. 2. If the point P have as coordinates x' and 3/'
then, since x' =^a cos <^ and 2/' = 5 sin <^, the equation to the
circle of curvature is

\^-^rn ^[y^-w-y) ^^-^•

L. 26



402 COORDINATE GEOMETRY.

Cor. 3. In a similar manner it may be shewn that the
equation to the circle of curvature at any point {x\ y') of

9 9

%)C II

the hyperbola — — — — 1 is



a



y ) - ^aW









424. If a circle and an ellipse intersect in four points,
the sum of their eccentric angles is equal to an even
multiple of tt. [Page 235, Ex. 18.]

If then the circle of curvature at a point P, whose
eccentric angle is 6, meet the curve again in Q^ whose
eccentric angle is ^, three of these four points coincide at
P, so that three of these eccentric angles are equal to 0,
whilst the fourth is equal to ^. We therefore have

3^ + <^ = an even multiple of tt =::: 2mr.

Hence, if <^ be supposed given, i.e. if Q be given, we

u ^ 2mr-cf>
have tf — o •

Giving n in succession the values 1, 2, and 3, we see

, , - T 27r - d> 4:7r-di Qtt - cj>

that 6 equals — ^— ^ , — ^ — , or — ^— .

Hence the circles of curvature at the points, whose

^ 27r — cf) 4:7r — d> , Gtt — </) „

eccentric angles are — - — , — ^ — » and — - — , all

pass through the point whose eccentric angle is cf).

Also since

27r — d) 47r — d) Gtt — d) , , ,,. , n

r + — —J- + — -— i- + ^ = 4r7r = an even multiple ot tt,

3 3 o

, - . , 27r— d) 4^7r — (b Gtt — d)

we see that the points — - — , — ^ — , — ^ — j and </>

all lie on a circle.



EVOLUTE OF A CURVE.



403



Hence through any point Q on an ellipse can be drawn
three circles which are the circles of curvature at three
points Pj, /*2> <^^^ -^3' -Also the four points Pj, Pg, P3, and
Q all lie on another circle.

425. E volute of a Curve. The locus of the
centres of curvature at different points of a curve is called
the evolute of the curve.



426. Evolute of the parabola 'if=A:ax.

Let (x, y) be the centre of curvature at the point (<x^^, 2a^)
of this curve.

Then ic = a (2 + Zf) and ^ = - "latK (Art. 422.)

.-. {x-2ay=21aH'=^^ay\

i.e. the locus of the centre of curvature is the curve

21ay^ = i{x~2ay.

This curve meets the axis of x in the point (20-, 0).

It also meets the parabola
where

27a2aj = (a;-2a)^
i. e. where x = ^a,
and therefore

y = ^4.J2a.

Hence it meets the parabola at
the points

(8a, ±4^2«).

The curve is called a semi-
cubical parabola and could be shewn
to be of the shape of the dotted curve in the figure.




427.



XT 11

Evolute of the ellipse — „ + ^„= 1.
^ a^ ¥



If {x, y) be the centre of curvature corresponding to the
point {ct, cos ^, b sin ^) of the ellipse, we have



a'-b^



x =



cos^ <fi and y = —



a'







sin^ <fi.



26—2



404



COORDINATE GEOMETRY.



Hence

{axf + (hyf = {a? - h'f {cos^ «/, + sin^ </>} = {a' - h'f.
Hence the locus of the point (x, y) is the curve

{axf-\-(hyf = {a?-h'f.
This curve could be shewn to
be of the shape shewn in the figure



where



CL = CL'



a^-¥



a



and CM^CM'^



a'




The equation to the evolute of
the hyperbola would be found to
be

{axf-{hyf=^{a? + h'f.

428. Contact of different orders. If two conies,
or curves, touch, i.e. have two coincident points in common
they are said to have contact of the first order. The
tangent to a conic therefore has contact of the first order
with it.

If two conies have three coincident points in common,
they are said to have contact of the second order. The
circle of curvature of a conic therefore has contact of the
second order with it.

If two conies have four coincident points in common,
they are said to have contact of the third order. No
conies, which are not coincident, can have more than four
coincident points ; for a conic is completely determined if
five points on it be given. Contact of the third order is
therefore all that two conies can have, and then they are
said to osculate one another.

Since a circle is completely determined when three
points on it are given we cannot, in general, obtain a circle
to have contact of a higher order than the second with a
given conic. The circle of curvature is therefore often called
the osculating circle.



CONTACT OF DIFFERENT ORDERS. 405

In general, one curve osculates another when it has the
highest possible order of contact with the second curve.

429. Equation to a conic osculating another conic.

If S—0 be the equation to a conic and T=0 the
tangent at any point of it, the conic S = \T^ passes through
four coincident points of S=0 at the point where T—0
touches it. (Art. 385, lY.)

Hence S= \T^ is the equation to the required osculating
conic.

Ex. The equation of any conic osculating the conic

ax^ + 2hxy + bif-2fy = (1)

at the origin is

ax^ + 2hxy + b7f-2fy + \y^=0 , (2).

For the tangent to (1) at the origin iay = 0.

If (2) be a parabola, we have h^ = a{b + 'K), so that its equation is

{ax + hy)^=2afy.

If (2) be a rectangular hyperbola, we have a + b + \ = 0, and the
equation to the osculating rectangular hyperbola is

a {x^ - 2/2) + 2hxy- 2fy = 0.

EXAMPLES. XLVIII.

1. If the normal at a point P of a parabola meet the directrix in
L, prove that the radius of curvature at P is equal to 2PL.

2. If ft and P2 be the radii of curvature at the ends of a focal
chord of the parabola, prove that

3. PQ is the common chord of the parabola and its centre of
curvature at P ; prove that the ordinate of Q is three times that of P,
and that the locus of the middle point of PQ is another parabola.

4. If p and p' be the radii of curvature at the ends, P and D, of
conjugate diameters of the ellipse, prove that

and that the locus of the middle point of the line joining the centres
of curvature at P and D is

{ax + hy)^ + (aa; - hy)^ = (a^ •



406 COORDINATE GEOMETRY. [Exs.

5. is the centre of curvature at any point of an ellipse, and Q
and R are the feet of the other normals drawn from ; prove that the

locus of the intersection of tangents at Q and B is -3+ -2 = 1» ^.nd

X y

that the line QR is a normal to the ellipse

x^ «2 a^j)^



a2^ &2-(a2_ft2)2-

6. If four normals be drawn to an ellipse from any point on the
evolute, prove that the locus of the centre of the rectangular hyperbola
through their feet is the curve



©*-(!)*-•



7. In general, prove that there are six points on an ellipse the


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Online LibraryS. L. (Sidney Luxton) LoneyThe elements of coordinate geometry → online text (page 24 of 26)