CONCEPT

CIRCLES
MAP Class XI
• A d rd• I< dcllncil as IOCU> Mull such polntt In r l•no whlcil • General <'<!Wltlon of circle Is glv<n by x1 , f , Zgx ♦ 2/y • c = 0
reimlns 01 cons1on1 dis1ano:,, t"rom • glwn facJ poin1. l'IJC<J
point is called c<nltt anJ the lixc'(f dislllncc (di.<tance from wllh cemre (C):(-g. -J) and radius (r) = ✓8' + f' -c
centre 10 cad, point oflocus) is coiled radius of the circle. • (x-<11)1 , (r-lW=n1 wllhcentre(C)=(cc.13)
• 1he s«ond dcgre• tq1111tlon ai' + /J)" + 2/vcy + 2gx + 2/y and mdlu.1 (r):11
+c = Orepresents 1he <'ql1.ltlon of circle. If a = b, /1 = 0 and • Shnples1 Touching Touching Toud1lng bc,1h
a It R x-axb )'•axis x&y-uis
6= h I, /-,,oandJf+f' -11c>0

~~
11 J C )
'

If th• 1wo clrdc, C1 ond

C2 IP''" by i' + + r U&MJIIOII
2g1x + 2f,y +c, = 0 anJ
i'+ )~+2gr•+ 2J,y+c2=0
r~ctively lnlersect at
C@hfillllli • Diamet ric Form:
(x-x1)(x -x,) • (y- y 1) (y- y,) where

•
a point Pund PA, PB ore '
die tangcnlS al I'. Ihm (x 1, y 1) and (x,, y1) are end points of
1hr angle bc1ween lhr diameter.
t,,ng,nis al P (I.e. LJJPA) • Paramelrlc Fonn:
Is the angle urln1crseu1~n AJ I yl, 2g;t, 2})', c:Oc:mbe
of th,· two circles glwn hy
r.' + r,' -(AB)' iiiliAAU rc:1\r<.1&tntcd u
xc - ~ i t cO\ O
~-

c.»O• 2 y J,r,inO
'i':
r ,/JI'+!'-,

• Le1S=.r2, /+2gxs 'lfy, c

s, =x.1 •>',' ; 2~,; 1/y,; c and T=;a, t YY, 1 g(.n x,) •f(r • r,> ''
- EqUJ>lion or chord of the circle S = o whose mid-point is ex,. y1) is 1' = S1•
- IA.'118tb ofin1cro:r1 made by 1J1c clrdcS=o withx-uxi. = ~ .
- E.q11i1tlo11 of 1an11"nl 10 the cln.lc S = 0 b given by '/' = 0.
Lcngll1 of1111erc.111 nwdc by lhc clrdc S" 0 wilh y -11.d, = 2,//' -< .
- 1.cnglh of •••sc111 from a rolnl (x1, y1) 10 the <lrdc S = 0 I• given hy .Js..
- ·1 be C'!U,tllon of r...uc.J uts uf IWOdrd<> s, "o. sl = 0 In 1<bkh lh< coofhdcnl or ...yl
Tuil Circle, aro orlhogonal ,,.. <iunc b I he <'<!U>llon lll°lhc line S1- s, = O.
If 0 = 90" • IJn•y= mx, r will touch 1heclrclc,.i, /=al If i'- = a1 {I t 111:lj.
i.e., r1'- + ri1-(1\Hfl:O • f.quauon o( direc.lor circle of the c:irclexl , i-; ,r is 1he drdc whose radius is ./?. limes
or 2g1g 2 • 2/J, = c, , ,, lhe radius of the origin.al clrdc.
 (Q,iNCEPT
!PARABOLA,
MAP Clas,s, XI

Standard Fi1n11
De:ftnlllan
~i\-ifar.J il.'11 lff ti l rar.1~16 l!11 ,:l = '4~ :i. U
P&rM)l,U it !, I.CIL\i "' F.,\,,,"l;b. ( "l I 'ia,, 0)
ul ~nl wli1c.:~
I • \ trl"'-' : (~11 11)
mrr,-n. ln III pl,uw
au, h lf1i1I ii"
111 lti,c-.lfLI (j l JI • -
1h•••n.:r Imm 11 ~ .\1.i- ) • If
hi~~ nl l, i;t111 ii
• I •~•-11~ O!IL ht't.el. Jt 0
I'll lb J hl>1llu!
flum • bk'll line.
---·► "' I Ill• 1,i:dmn. f f "l!; T = 11
I.

"11'~ ti,AI rn 1111

h 11K ~ Ila
ji-...-"d lux b 1hc
1l11 t1,,•""
CNdHfDn II TM ttne, DI I ILl■1 1ta lhl hllllotl

,= -u
T "'
.um ,tlJ l )' •m K ■ p [' • - Ill l&rl U WI Lil
hinl U1_1UllJln1r11i1Tanp lL • ~.,,> ■ r1 t)
n, .2.11, • 1 11
U',nlnl lrnm)
•1• .1 • JIJ Sou lmparti1l laulll
(.P. !iin
fi't.i:r11m1. lrk 11nm)
"Y
(;.~)
[dllf"C n. _»n,i.111}
1"'"1lld - "
I'll
(S!ui•~ (mm]
,. Lxu., ,r,f .,.,rn1 ai~~m al rnrmu.J.nl1111r
I~ r r,1b11~ h, I• ~ ...ttL~. I ■ - 0
~.. l••

• Eq~''"'" . .~ cl~J I J f' it'iilia.b l,biii.l&'d ■I ~nl

..iiT1I l"llll'hfl hf'"• ~iJ(l " , ,, . , ,, _ ,,,J\ I
&iullin o:r Naraal ,. 'fll,lllr,11 ol d,o,J JI i,;11n1u.-.1

liLJ~l ,,r I.111,1;mh lo i11 puwbul1 .

T - Tu(ll~D nl ~ b Jtl'!.."11 In
Cane Pahl Condilin » 1• :!ti(,
,, • " ,)
- 11 t)l,J1 1l', ~ }. flh.irl , :l.i.'itl
--=--~1-~ - ~ J
.
,. - , , ...2J_ [L - ;a:}
~ ,ln:.11 ,•,1uulJtlO o! (JU 1.. .ih~t h• 1 >11 I
(I~ p:,lru r,om~
h) ,-~f 1 I r.,) ~ • 2,;l1 1i,
f • 'in I th • ,' I ~r
• 1fl;i'" fral,[1th N .. bu1J o, Iti..- r~t11h.~ _,J • lu.1 J~•1i..-lffil •bril\~li
(Pi1.11tiQd1li. (i I 110]
I~ nT1i..'1 ~mr nui.m;g •n ~Ilg.It" 0 k""'llh 1h.;: OIi I•"'~•°" ff
J ;nu - ?um - :J,,.-,
(l\mlt Jure iiuml
CONCEPT
ELLIPSE
MAP Class XI
.i:.t1
, Al (x1,y1): - +-YY12 : I
Definition a2 b
At(a cos 0, bsin 0): ! cos O+ l.sin 0: I
Locus of• point which
movos in such a my that ·
th< """ orlb dlst•ne<s
rrom I WO Ox<cl points i,
') Tangent
Equotion of
tongcnt Is
gh'Cll by
lns.101,~(om, : y • mx± Jn2m2 +62
"

• Equation of ~roftangenl S," S 11S, whcro

b

xl .2 < X •• • xl li.l

-..
alwaysconst.:1n1. S • - + ' - - 1 S ._ .::.L::.+L.tL. - l S .:!L+ - I
"l bl . t n2 bl , 11 .,2 I}
• Equation of chord or contact of 1>n1,..,nto dro"'n from n

-
Condition of
J!llipu
point (x1,y1) to dlipse i< ;

Equation of lint
1-; m.x+e
11·
+ ~• : I
b

Condi lion

tangency for
a line to the xcosa+yslna ~p P' s 111cos'a + li'sin1u
ellipse

- Equation

y
(0. b)

b/ f ~
Standard Form
(0. P(x.y)
x2 ;
- + - :c: l
c.,. ,1•)1'
<-:--"1-.oi c... oN.. 0> ;;; o
"1 b'! Graph X' . '--- (0, O) _ / • X x··- ,-.
... - O,+
) +"~"'"'""<.."""o"")-+X

- - (0, -lo)

I" \ -(o'.'_b)
Y'
CC!nlre (0, 0) (0. 0)
Normal
Vertices (ta. 0 ) (D, ±b)
l~quouion of
nonnal LA,ngth ofmajor llis 2a 2b
is givi,n by
t..,ngth or minor axis lb
Foci (:l:ar, 0) (0, ±b,)

• At(.r 1,y1) :
'
a2--
x b- 2 y
.rt Yt
• At(11 cos 0, b sin 0) :
• 11l- /;l
Equation of dlrtetrlc..

Ecc•mrldty ,.g
u.ke 0-bycos«O c al-b'
n1(a2 -b2)
• SloP<'form:y:m.r- ~,...._ _..
Lt,ngth oflotus r<etum
2a2
.Ja2 +b2,,,2 a b
 CONCEPT
HYPERBOLA
MAP Class XI
• Locus of a point which moves in such y
x2 y2
a way th at the differe nce of its distance Standard for m is given by 2 - 2 =1
fro m two fixed po ints (foci) is always a b
constant.
• General equation of second degree
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
a h g
represents a hyperbola if ~ =h b f -# 0 and h2 > ab.
g f C • Centre: 0(0, 0)
• Foci : S(ae, 0), S/- ae, 0)
• Vertices: A(a, 0), A/-a, 0)

•
. .
D1.rectrices: x =-a an d x = - -a
A hyperbola is said to be : Definition e e
rectangular hyperbola, if the : • Transverse axis : AA 1 = 2a
asymptotes are at right angle : Conjugate axis= 2b b
I 2 2
i.e., 8 = 90°. Equation of : • Latusrectum: LL1 =L'L{ = - = 2a(e 2 - 1)
a
rectangular hyperbola is x 2 - y2 :I
= a2 or xy = c2 • :
·---------------------------------------
~
--------------------------· Standard
form
Rectangular
I I
Definition Hyperbola I
I X Y
2 2 I
I
1 If -
I 2 - -2 =1
is one hyperbola> 1I
•I a b •I
Properties : the n its conjugate hyperbola is :
: y2 x2 :
• given
I . by - - - = 1 •I
Important : b2 a2 :
Results

If +ti> ~ } nd ct B( 2 .:, ) are the

2 2
po ints lying on the hyperbola xy = c2 then X Y X1X Y1Y
S = - - - -1,S1 = - - - -1,
• Vertices of xy = c2 are (c, c), (-c, c) a2 b2 a2 b2
2 2 2 2
• Foci : (± c✓2 , ± c✓2 ) su -=1_21...._
a2 b2
1 s =X2X _Y2Y -1 s =X2 _Y2 -1 s =X1X2 _Y1Y2 _1
, 2 - a2 b2 , 22 - a2 b2 , 12 - a2 b2 ,
• Equation of directrices is
x+ y±c✓2 = 0 • Location of P(xl' y1 ) : Pis inside the hyperbola if S11 > 0.
P is outside hyperbola if S11 < 0, P is on the hyperbola if S11 = 0.
1
• Slope of chord at A =-- • Chord with m idpoint P(xl' y 1) : S1 = S11
tl • Chord joining the points P(xl' y 1) and Q(x 2, y) : S1 + S2 = S12
• Chord subtend right angle at the • Tangent at P(xl' y1) : S1 = 0
vertices. • Chord of contact of tangents from P(xl' y 1) : S1 = 0
1
• Slope of tangent at A =-2 • The tangents with slope m are y = mx ± .Ja2m 2 - b2
t1
• Locus of the point of intersection of .
• Pair of tangents from P(xi> y1) : Si = S
11 S
I
2 2
tangents at the end point of chords is : a x b y
I • Normal at P(xl' y 1) : - + - = a2 + b2 = a2e2
x + y= 0. : X1 Y1