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639 views16 pagesLocus
Uploaded by
Rachit JainCopyright
© © All Rights Reserved
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/ 16
CONCEPTS AND OBJECTIVES IN
MATHEMATICS
VOLUME -1B : PART-1
(GEOMETRY)
VARSITY®
EMPOWERING EDUCATION * ENABLING DREAMS
Central Office Working Office
4° Floor, Plt # 80, Sti Sai Plaza, DNo, $4-20/9-6, Floor 1,Timmarusu Stree,
Ayyappa Society, ‘Srinagar Colony, VIJAYAWADA - 520 008.
‘Madhapur, HYDERABAD - 500 081. Ph: 0866-2544369
Sri Bhanu Kumar, A., Lecturer in Mathematics,
Sri Chaitanya Educational Institutions, Hyderabad,
Dr. Robinson Edward Ri Professor of Mathematics,
Indian Statistical Institute, Bangalore.
Sri Srinivasa Rao, K., Academic Dean,
Sri Chaitanya Educational Institutions, Vijayawada,
Printed at :
‘SREE VIJAYA GRAPHICS
# 43-106-1/1228, Kanaka Durga Nagar
\VWAYAWADA - 520 015, Ph : 0866-2402256 CONCEPTS AND OBJECTIVES IN
MATHEMATICS
VOLUME - IB : PART-1 (GEOMETRY)
(Sumber of pa
© VEML, Hyderabad Price : R
All rights reserved (etusve oa are)
Fist Edition: March 2012
Revised Edition: March 2020
7 CONTENTS
2D-GEOMETRY
Co-Ordinate System
Locus
Change of Axes
Straight Lines
Pair of Straight lines
3D-CoOrdinate System
Direction Cosines & Ratios
Planes Locus
ition : The set ofall points (and only those
s) which satisfy the given geometrical
condition(s) (or properties) is called a locus,
From the definition ofthe locus, it follows that
{every point satisfying the given condition(s)is,
‘apointon the locus.
fi) every point on the locus satisfies the given
condition(s),
i) The set of points in a plane which are
‘equidistant from two given points A and
is alocus, Here the locus i a straight line,
In otherwords, the locus of & po
equidistant from two given points A and B
is {P/AP = BP). This set represents
the perpendicular bisector of the line
bs
segment AB,
é y 8
Pat
ii) Let and B be two fixed points. Then the
setof points P staistying ZAPB =90? isa
locus. Here the locus isa circle.
The locus of Pis (P/ZAPB= 9}
[2a }e
© Parametric equation of locus
LOCUS
@ Locus
Equation of a locus
e whi
i) The set of points in a pla
constant distance r from a given point Cis,
locus, Here the locusis a circle
‘The locus ofa point P whose distance from
a fixed point Cis r units, is (P/CP =r}.
Locus
Figs 133
If there are two or more geometric conditions
describing a locus, these contions mst be consistent in
the sens hatter existstleast one point satsying these
conditions. Others, there exists no locus,
EQUATION OF A LOCUS
An equation fix, y) =Ois said tobe the equation
of a locus Sif every point of S satisfies fx, y)=0
and every point that satisfies f(x,y)= O belongs wo.
12
‘An equation of a locus is an algebr
description of the locus. This can be obtained inthe
following way
i) Consider point P(x, y) on the locus
fi) Write the geometric condition(s) to be satisfied
by P in terms of an equation or inequation in
symbols,
i) Apply the proper formula of coordinate
geometry and translate the geomet
condition(s) into an algebraic equation
iv) Simplify the equation so that itis free from
radicals,
‘The equation thus obtained isthe required
equation of locus,
[NEO SERIES for Sri Chaitanya Jr. NEON Students] af SOLVED EXAMPLES
Find the locus of the point which is atm constant
distance of S units from (4-3).
L LetA=(4,-3)and Ps, ») bea point ond
Given condition i, PL= 5
36-9043"
Find the equation of acu ofthe pot for which the
‘sum ofthe squares of distances from the coordinate
aes is 25,
{Let Pur, 9) be a poi in he Hous
Distances fom Po the coordinate aes ate ||
Given condition is [yf +f
Iecus of Piss? y°=25
he distances from P tothe points (3,4) and (3,4)
arein the ratio :2,findthelcus of?
Let Ps») be a pot in the ls
Lad=G.4,8203.4)
5 ra
Given condition is £4
ss =9Pn?
alles) 4-47 $937 +0-4)]
5084592478 doyt1a5=0
s locusof PS $2 45y2-478%=40y +1250
AQ,3).81,5),C-1,2arethree points Pisa pit
ehh 4+ PH? =27C Ahead teas of
Let Poy) be any point on the es.
Given condition is PA? + Pa = PC
BHO 3F HO DEH 5?
=a(oriF +(r-2))]
=U s1edee teddy)
3 625 6r— 16439
= lors 8y-29=0
2 The locus of Pi Ws +829 0.
e{Locus]
‘The ends of the hypotenuse ofa right ange triangle
are(06)and (6,0) Find the locus of the ted vertex.
(MARCH.2008)
Let A(0, 6), B16, 0) be the ends ofthe hypotenuse and
Py) be the third verte.
Given condition is ZAPB = 90°
PPAR aA: PAA PAB
3 P40 OF +0 OF 726
PSPS dpe 364 126636
3 28 42y-12e- y=0
3 P4 P64 69=05 (4.99 40,6) 0F(6,0)
4 The hous of Pis x + 9°66
where (3) 410.6) 06 (6.0)
ACL. 1) 81-2, 3) are two points. Ifa point P forms a
‘wiangle ofarea2square units with A B then find the
locus of.
Let At), 1), (-2,3) the given pins. Let Xs, ype
pola othe lous
Given condition ig Area of 4 PUB =2
we eytse2-ayefas
xntyt shaban (2emayes?
$5 Ae 1294 97220430549
4. Te lous of Psd 129 4952 205-205 4920
AUS, 3), BC, -2, C12, 1 are thee pats IEP
point such tha the area of the quadrilateral PACs
square units, then find the locus oP.
La P=( any point on Iocus
Given condition is
are ofthe gusdrilsteral PAC = 10s, unit.
S824 99-40-99 [=20
=}6-3y
3 |4x—39—18]=20 49 —3y-189 = 400
162 —244y 4 92 144+ 108-750,
2 The locus of Pis
16x? — Day +992 — bes toy
(NEO SERIES for Sri Chaitanya Jr. NEON Students jf 25] 8. An iron rod of length 2 is sliding on two mutually
Perpendicular ines, Find the locus of the midpoint af
therod.
Let the two mutually perpendicular ines be
coordinate axes
Let A(a0), 0, bye the ends the fod. Let Ps yb
the midpoint of AB
40 sy
The los of Pins
A straight rod of leg 9 units des with ts nds
A, Balwaysoon the x and yeases respectively. Then
centro of 4 0AB.
00,
Given AB =9 = a 45a 81
Let Ps,» the centroid of AOAB.
2 The locas of Pis +9? =9.
VERY SHORT ANSWER QUESTIONS
Find the equation othe locus of the point which
is atadistance of S units from (-2, 3).
Find the equation o the locus ofthe point, for
Which the square of whose distance from origin
is 4 timesits y-coordinat.
<(MATHS.8, PART
Find the equation to the locus of points
equidistant from the poin
D (3.20.4)
ii) (a+ b,a—b).(a—b,a +b)
Find the locus ofthe point which is equidistan
from the coordinate axes,
'). Find the equation of locus of the point
‘whose distance from the coordinate axes
are in the ratio 2: 3
ii) Find the equation of locus ofthe point which
is ata distance Sunits from the Y-anis,
ii) Find the equation to the locus of a point P
whose distance to (2, 0) is equal to its
distance from y-axis,
Find the locus of P for which the distance
from Po origin is double the distance from
Po the point (1,2),
(TUNE-2005) (MARCH.2012)
ind the locus of Pif the distance of P from
3.0) is twice the distance of P from (-3, 0),
[SHORT ANSWER QUESTIONS
the distances from P wo the points
(2,3), 2,3) are i the ratio 2:3, then find
the locus of P.(\IAY:2014) (TS MAYS)
(5,4), (7,6) are in the ratio 2:3, then find
the locus of P, (MAY:2008, March 2014)
Find the locus of the point P such that
PA? + PB? = 2 where Aa, 0), Ba, 0)
and 0< a < |e] (PMY 18)
AG, 2), BQ, -3), C(-2, 3) are three
points. A point P moves such that
PA? + PB®=2PC2, Show that the locus of
PisTx-Ty+4=0 (Mtay:2007)
Find the locus of a point P if the join ofthe
points (2, 3) and (-1, 5) sublends aright
angleat P. (MARCH-2005) (MAY:2012),
Find the locus ofthe tind vertex of aright
angled triangle, the ends of whose
hypotenuse are (4, 0) and (0,4)
[25 _______________-¢[NEO SERIES for Sti Chaitanya Jr. NEON Students] «{tocus]
yB=0 Gn as Gabe IS)
ii) P4924 Ay=0; (#0, Hor (4,0)
3), BU-3,4) are two points. Ifa point [yep
ch that the area of PAB is
8.5 square units, then find the locus of P.
P moves s
‘atamenianiy [& 9 8 100°4259°—34e—1709=0
i) A(S,3),83,-2)aretwo points apoiatP | i) (x-2)-s)6r-2y—-D=0
forms tangle of area9 square units with Jey ayes
A. Bihen find the oeus of P
(MARCH.2009,2006) TS MARIS) Jo, iy 42 i) 162
5. i) 0(0,0), 46,0) (0,4 are thre points. 2
If Ps a point such that the arca of APOB
is twice the area of APOA, then find
the locus of P, ADDITIONAL INFORMATION.
Find the equation of locus of the point [1-3] PARAMETRIC EQUATION OF A]
‘which i collinear with the points (3,4) and Locus
>,
While defining the property obeyed by the
6. i) Find the equation of locus ofa point such point ofthe locus, sometimes.x and y coordinates of
that the sum of whose distances from (0.2) any point are given as separate functions of a
and (0,2) is 6. variable,
i) Find the locus of P,ifA = 2,3), B= (2-3)
and PA + PB=8 (MARCH-2008)AP MAR 15)
“
Let x= f(@)and y = 9(8). Now for all
admissible values of @ € R the point x,y) defines
7. id) Find the equation of locus of apointsuch that 4 Jocus. Thus the relation between x and y is
the difference of whose distances from
(-5.0)and (5,0)is8 (MAV:2006, 2011)
Fd the equationctlocaset P.if4= (40), The variable @ is called the parameter and
indirectly conditioned by the variable @
Cin) ad eee 5 =/@), y = 9(0) are called the parametric
ak eeenwavansy efwatons ofthe lous.
ANSWERS} 14] CARTESTAN EQUATION OF A LOCUS
VERY SHORTANSWEA QUESTIONS FROM PARAMETRIC EQUATIONS
heey tro—12 Lette parametric equations ofthe lous be
b, s24)2-ays0 1 =Adjand y= 6(). tis parameter.
BD reay ‘The Cartesian equation of a locus is a direct
relation between randy. Forth the parameter 1
qi is eliminated fom te tworelations anda fanctonal
. oy
‘8 ADDITIONAL SOLVED EXAMPLES }'t:
i) Pays lors 980
1. Find the Cartesian equation of the lacus whose
‘SHORTANSWER. QUESTIONS parametric equations are x =a cos0, =a sino
20c- Tay +65=0 where @ isthe parame
iy 5245)? 34n+ 120942 Sok ey
(acon )*+ (using =o
* The Cartesian equation
[NEO SERIES for SH Chaltar IEON Students 2 Find the oeusof the point a bsee, ¢—btam 9).
where @ isthe parameter.
Sol. Let Ps.» be the given pon
Nowr=a-bsec@ = r-u=—bseco
yeanbung@ = y-a=—bung
seo 0 un? = 8°
uation the lous is (4 =)? ab
Fld the locus ofthe point a cas? asin! 0).
where @ isthe parameter.
Sol. Let(x.») =a con°®,asin?8)
=
and adding
sind
*. Equation tothe lacus ie +97
4.0) Find thetocusot the point
ens),
1) Find thetocusof point
(tang sin 9 tam 9-sin where 0 isa parameter
Sol. i) Let{coseed —sinO, sec ~cost = (x9)
sn
in eo
snd “* 80 =?
“sin “cos
and ay =we
= cos sin?
Fy say? =H
8) Let(un 9 +sin0,tn9-sing) =e»)
tan +sin @ = rand wn —sin @ =y
2 pasno
une
‘ADDITIONAL EXERCISE
Find the locus of the point (x, ) where
i) x=a+beosg.y=b+asing
ii) x= a+ bsecg,y =b+atanB
iil) x= sec@ +btan@,y =a and + b sec
for all admissbile values of 8, @ is a
parameter.
Find the locus of the point (a cos*® . a sin*® ),
where @ isa parameter,
i) Find the locus of point (ar®,2an) where ris
parameter
(_<
i Find he oes of pin ner isa
parameter, .
where fis a parameter
3 Fete
1 isa parameter
TANSWERS}.
ANSWERS
o-)
ar
(mat
(=a?
Vir feede
a) year
9 Wad) sab=0
[28 __________¢[NEO SERIES for Sti Chaitanya Jr. NEON Students] (waTHs48, PART-1}®
(SYNOPSIS
‘The pat traced out by moving pot under one
or more given comiions is called its “Locus
‘he sigetraic relation between xan y obtained
by applying the geometrical conditions is called
the equation of toes.
“The locus of point whichis equidistant fom
the two points AC) and Bs, 9) the pe
pendicular bisector of AB. 1's equation i
The locus of a point which is at a distance of “P
units from the given point ¢x,.y,) is circle
whose equation is (s=5,)' (y= y)F =e
Let Aix, yp), Bet ¥,) be two fixed points. The
locus of P such that ZAPB = 907 is a circle on
the line joining A, B as the ends of a diameter
and its equation is (x1 )0-3,)4(949,)0-9)=0
‘The locus of a point “P” such that PA = KPB is
1 perpendicular bisector of AB if K = 1
ii) acitcle if K#1 and k>0
i) empty se if <0
ILA, ate wo points then the locus of point
such that PA? + PB
ne if K
ii) acitcleif K #2 and K>0
ii) empty set if k-< 0
Let A, B be two points, then the locus of P such
that the area of triangle PAB is A isa pair of
Parallel! ines which are parallel to and at a
distance of 24 from AB
Let A, he two poins, then the locus of point
‘P such that PA + PB = Kis
1) amollipse if K > AB
ii) a line segment if K = AB
iy empty set it K < AB
(Locus)
Let, B be wo points then the locus of point
P such that [PA~ Py
3) alypesbola if K < AB
ii) union of two rays i K = AB
i emmy set if K > AB
1A =, by B= a,b) then the lous of P such
that PA + PB =k or PA = PB
42 40-0?
ee aa
1A = (a, by B= a,b) then the lous of P such
that PA + PM = kof PA = PR = Kis
sea) ay
Boab
The curve represented by
Saas! +s? +2hwy+2—x42
Aa abe+2fehmaf? be! ~c
|) acileifa ed, had, s° +/? —ac20.A70
ii) apsiroflinesit AO, # 2ahe? 2a, 7? 2b
i) a pair of parattines if
0,1? = ab, af
iv) a parabola if a #0, 1?
¥) Ancllipse if A 0, < ab
vi) a hyperbola if A0, J? > ab
ia recungular hyperbola if 820, a+
and > ab
‘The equation of the locus of the point whose
distance from «avis is twice its distance from
the yeaxis is
pyaae Day
Dyade Hartye0
‘The locus of a point whose distance from the
y-axis is half of its distance from origin
‘The locus of the point, for which the sum of
the squares of distances from the coordinate
[NEO SERIES for Sri Chaitanya Jr. NEON Students |: 4. Ifthe equation to the locus of points equidis-
int from the po ) is
axtbysea0) where @ > 0 then, the ascending
order of a,b, € is
Habe Doba Hhoa Nach
5. The locus of the point which is equidistant to
the coordinate axes is
Dey
yee
ae
6. If the equation of the locus of a point
equidistance from the points (a, b,) and
(ay by) Is (a,— ashe+(b,~ bylyse = O then the
value ofc is
+b by
3 aja} +)
4) Mal +a} 458 +83)
7. ‘The point P moves such that the sum of the
squares of its distances from two fixed points
Ala, 0), B(-a, 0) is 8a the Hocus of P is
8. ‘The equation to the locus of P such that the
Join of (a,b) and (b, a) subtend a right angle
ats
tty ax~ dy +ab=0
2)x8+ yar —by + 2ub=0
Hey (a+ Det yy eab=0
D4 C@ + DEY) 420-0
9. Aline passes through a fived point A(a, 6).
‘The locus of the foot of the perpendicular on
it from origin is
Dees tae by
10. The locus of a point which is collinear with
‘the points (3, 4) andl (4, 3) is
1 2643y-12=0 2) x43y412.
0
3) ar42y412=0
4) x-Ty425=0
nL
2
13.
14,
16.
NEO SERIES for
{MATHS-1, PART-
IA = (6,0), B = (0, 4) and O is the origin,
then the locus of P such that the area of
APOR » 2area of APOA) is
2 3y
be > 2tedy
0° 4a ay?
‘The locus of P such that the area of APAB is
12 sq, units where A(2, 2) and BC, §) is
1) 24 xy + 992 224 — aby = 28
2) 2 Gay +9 + 228 + 66y 4.23
3) 24 Say 4 92 + 20 + 6y + 2
4yx1 dy? dy 12 —24y
‘The base of a triangle lies along the line x=a
and is of length a. The area of the tangle is
«2, The locus of the third vertex is.
Do +a)e-m=0
2x a) +30)
3)(x-@ e320
(ea) 2a) =0
AUS, 3), BB, -2), C2, -1) are the points and
P is 4 point such that the area of the
‘quadrilateral PABC is 1 sq. units then the locus
of Pis
1) 162-245y49y?-14444108)-7600
2) 6r2424ay49924 144410894 76=0
3) N6x2424uy49)2 144410897
4) 1ox2e24ay49)%41444108)-7620
A
‘om the axes,
ye segment of length 2! sliding with ends
‘the locus of the middle point
of the Tine segment is
4P
=4P
4x
pes
ae
A straight rod of length MV unit slides with its
ends A, B always on the x and y axes
respectively. ‘Then the locus of the centroid
of AOAB
# QeeyeP
aeteyt eae
atey=2e
haitanya Jr. NEON Students] MATHSB, PART [2
17. Through (4g, y) variable line is drawn cut
ting the axes at A, B. If OACB is a rectangle
then locus of C is
18 Ith wean, k denote
G.M. of the intereepts made on axes by the
lines passing through (1, 1) then (A, &) Hes on
by ae
yar
te the arithmetic
Ayn + y=2sy
19. Ia, b,c arein AP, a, x, bare in GP. and
», ye are in GP. the point (ry) les on
Dt ey a2 29? 22H
2s
3) tay =H 4) tay emt
20. Let A = (2, 5) and B = (4,-1) are two vertices
of a AARC. Third vertex © moves along
L=9x + Ty +4-=0. The locus of the centroid
of triangle ABC is the line
1) 96-7y-22m0 2) Or47TV422—0
3) 2e421y-770 4) 274219780
24. A point moves in the xy-plane such that th sum
of its, two mutually
perpendicular lines is always equal to 5 units.
‘The area (in square units) enclosed by the lacus
distances from
‘of the point i
25 .
D> p23 950 4 100
22. Ala, 0), B(-a, 0) and ZAPB= 45°, then the
locus of P is
Dress arta
pete
Dey 4 day ee
Dat ey? — day!
A= (4,0), B= (a, 0).P isa moving point such
that ZPAB— ZPBA
The locus of Pis
2
9 e-¥
a» x 5
(NEO SERIES for Sri Chaitanya Jr. NEON Students
(Locus
24. ‘The locus of a point such that the sum of its
distances from the points (0, 2) and (0, -2) is
68
pox? — sy? 45,
3) 9 4 Sy? 24s
25. A point P moves so that the sum ofits distance
from the points (ae, 0), (-ae, 0) is 2a. The
Tocus of the point P is (0 )
A\ point moves such that the sum ofthe squares
the sides of a square of
10. 2 passing through the origin
cus twoparalle line x~y +10 = 0 and x —y +
20= Oat two points A and B. IPP is « point on
the line *L? such that OA, OP, OB are in
then the locus of P
harmonic progression,
1 ac4ay 44020 2)ar4 ay 420
0
0
3) Re Fy H4D=0 4) Rey 420
WA
ine drawn through a fixed point (, ) cuts
the axes at and Q. The rectangle OPRQ is
completed then the locus of R is
nied
By xtyshek
4) x-yehok
12, Ifthe point “P* is equidistance fro
AQ, 3), BUR, 8) and C15, -1) then P
bs 088 4 svio
{33)
3) 28 13. The base ofa triangle lies along x=1 and is of
length 1. The area of triangle is 1. The locus
of vertex is
De+De-3)=0
Der
dire d=0
Dut Es3=0
HEHE =O
14, ‘Two straight rods of lengths 24 and 2 move
long the coordinate axes in such a way 1
their extremities are always
the lacus of the centre af such circles is
Ue yaw ee
2) 268 -yyaa eh
19.
15, ‘The set of all points that forms a triangle of
‘area 15 sq- units with the points (1, -2) and
(5.3) lies on
1) Sxs6y423
2) (5x+6)-23)(51+6y +37)
20,
3) 25x +36y" +244
30y-227
4) Se+6y—37
16. ‘The locus of al points that are at distance
of aeast 2 units from (80) is
» oxy fey 4604750}
2 {(x.y)}s' +» +6x452 0}
9 {(xs)}e +y* ox seo}
4) {(uy))e + 9° -6r+5s0}
17, Mate the oct
a
of the points in List-I with
the curves in List-IT. (here pg and r are
‘constants and 1,0 and are parameters)
(sa)
lic, Then 18.
NEO SERIES for
,
By (p+g00s6,r+qsinB) I) circle
© (4h 11) ctipse
19) hyperbola
(TS Fameet 2019)
2) Ad BAL CA
4) Ad, Be IM, CV
1 AAV, BA, Cal
3) AAV, Bul, CA
‘The ends of a rod of length F moves on two
‘mutually perpendicular ines. The locus of the
csi in the ratio
point om the rod which di
1
1) 3x +4y
3) 4x +3y
2P 2) 9x +16y" =P
=6F 4) 98436)" =4P
A point p moves om the line 2x —y + 4=0. IP
O11, 4) and K(3,~2) are fixed points, then the
locus of the centroid of APOR is a line
Mel to x axis
DP
2) with stope 2
3) with slope = 4) parallet to y axis,
mnces from a variable
points A(,0) and B01)
(AP Famecet-2019)
2) 16x? + Ty? —64e—48y
3) 3e 4x-dy=0
DIG +38xy+7y? —4x—48y =0
42ay+3y
A straight line meets the X and ¥ axes at the
points A, B respectively. If AB=6 units, then
the locus of the point P which divides the line
segment AB such that AP:PB=2:1 is
Hav+y=36 2» 4r+y'=36
Hav ey 16 yd ty =16
haitanya Jr. NEON Students] MATHSB, PART
such that the distance from a point P to the
line BC is equal to the geometric mean of the
istances from P to the lines AB and AC, then
the point Plies
Dey tay
he curve
°
28s 4 By-8=0
HAs ede + THO
Ay 8E- TE + Dy 8
°
23. A line moves such that the portion of
intercepted between the coordinate axes is of
‘constant length a, then the locus of the mid
point of that tine segment is
4 veya
grey ee
4
24. The locus of point P(x, y) satisfying the
equation Yor—2F +9 + Jee BF ty 4,
is
1) an ellipse
2) aparabota
3) a line sepme
4) a circle
25, For any value of @ ifthe straight lines
-xsin0 + (1~e0s®)y =asin® and
xsin0—(L-+cosO)y +asinO=0 interseet at
P(O), then the locus P(Q)is a
1) Stexight line 2) cincle
3) parabola 4) hyperbola
26. The locus given by
Var ~ 2Any + 99"
1) pair of lines
Oar + My +46 =0 is
2) cincle
3) parabola 4) Hyperbota
(NEO SERIES for Sri Chaitanya Jr. NEON Student:
s{ Locus]
Second year Problems
27, ‘The equation x24y?-2e-4y-Sa represents
1) circle 2) pair of straight lines
3) ellipse 4) a point
28. The locus represented by the equation
Bede tty 8205s
1) hyperbola 2) circle
3) ellipse 4) parabola
29, ‘The locus given by y+ ty -1= is
D circle 2) hyperbola
3) ellipse 4) pair of Hines
30. ‘The locus of the point (a eosé x, b sive x
} Cirle 2) Ellipse
3) Hyperbola 4) Parabola
31. A and B are fixed points. If |PA—PB| = K
(constant) and AB then the locus of P is
1) Hyperbola
3) Parabo
2) an ellipse
4) a circle
33. The locus represented by the equation
Deb 4 day —2yh 4 de ay
1) A reotangul
hyperbola
2) a parabola
3) An ellipse 4) pair of tines
34. Ifthe distance of any point p(x, y) from the
4(P,Q)= max {}x—x\,|y~y)|} If Q is Fixed
point (1, 2) and d(®, Q)=3 then the locus of P
2) straight tine
da
jangle
[35] (Locus
Tanswens}
‘ANSWERS
122 31 43 3
63 73 82 91 103
12 124 131 14a 152
1692 173 18)4 192 203
21)4 22) 293 -24)3° 25/2
26)3 27/1 28)4 292 90)3
311 92)2 39)1 343
(EXERCISE -
and b,¥,.¥55 form two,
Infinite A.P's with common difference p and q
respectively then the locus of Put, &) when
ke
D ge- a) = yb)
2 plea) = gy)
3) ple ta) = giv D)
4 ply tay = qtr +d)
2. It the roots of the equation
(of