0% found this document useful (0 votes)
69 views16 pagesCircles 1
T
Uploaded by
mbrvsriram22Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
0% found this document useful (0 votes)
69 views16 pagesCircles 1
T
Uploaded by
mbrvsriram22Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 16
EXERCISE-I
EQUATION OF A CIRCLE USING
CENTRE, RADIUSAND STANDARD
FORM
Equation of circle passing through
non-collinear points A, B, Cis
1)Equation of the circle on AB as
diameter+ K (equation ofAB)=0
2) Equation of the circle on AB as
diameter+ K (equation of BC) =0
3) Equation of the circle on AB as
diameter+ K (equation of CA) =0
4) Equation of the circle on BC as
diametert K (equation of AC) = 0
The equation of the circle with radius 3
and centre as the point of intersection
,2x—y=]is
of the lines 2x+3y
D)xrty=9
2) xe +y?-2x-2y-7=0
3)x? +? -2x-2y+7=0
4)x7+y?49=0
Ifthe centroid of an equilateral triangle is
(1,1) and one of its vertices is (-1,2) then,
equation of its circum circle is
1) x? +y?-2x-2y-3=0
2) x+y? +2x-2y-3=0
3x +y?-4x-6y+9=0
De ayexyt5=0
Origin is the centre of circle passing
through the vertices of an equilateral
triangle whose median is of length 3a then
equation of the circl
2) P+y? =2a*
3)xt4y2=3a 4) xt yt =da?
The diameters of a circle are along 2x+y-
7=0 and x+3y-11=0. Then the equation of
this circle which also passes through
6, 7)is
1) x? +y?-4x-6y-16=0
2) P+ y?-4x-6y-20=0
Dey aa
1
i.
13.
a
3) xt +y?-4x-6y-12=0
4) x+y? +4x+6y-12=0
Accircle has radius 3 units and its centrer
lies on the line y=x-1. If it passes
through the point (7,3), its equation is
Itty? —4x-6y+4=0
2) x? +y?-14x-12y-76=0
3) x? +y? -8x-6y +136=0
4) x+y? -8x-6y+16=0
Number of circles drawn through two
points is
1)One 2)Two 3) Three 4) Infinite
If three lines are not concurrent and no two
of them are parallel, number of circles
drawn touching all the three lines
1) 122) 4 3)3 4) Infinite
For the circle ax? +y* +ox+dy+2=0
center is (1,2) then 2b+3d=
The radius of the circle passing through
(6, 2) and the equations of two normals
for the circle are x + y= 6 and x+2y = 4 is
DVS 2) 2N5 3) 3V5_ 4) V5
Centre and radius of the circle with
segment of the line x+y=1 cut off by
coordinate axes as diameter is
vGake aba
oltedhe ota)
. If a circle passes through the point
(0,0),(0,2)(0,5) then the coordinates of
its centre is
1) (a,b) 2) (-a,-6)
(: b -a ~b
3) $4) 4 (F 3)
The abscissae of two points A and B are
the roots of the equation x? +2ax—b? =0
and their ordinates are the roots of the
equation y? +2 py—q’ =0 then the radius of the circle with AB as diameter
is
1 feeerp ag 2 a+?
3) erg 4) Ja? +b? p-q*
Number of circles touching all the lines
x+y-1=0,x-y-1=0 and y+1=0 is
Number of circles touching all the lines
x-2y+1=0,2x+y+3=0 and 4x-8y+3:
The equation of the image of the circle
x4+y?-6x-4y+12=0 by the mirror
xty-1=0 is
Dxtty?+2x44y+4=0
2) xP +y?-2x+4y4+4=0
3) x°+y?+2x+4y-4=0
4) x24? +2x-4y44=0
17. The cirele concentric with
x +y'+4x4+6y+3=0 and radius 2 is
1) x+y? +4x+6y-9=0
2) x+y? +4x4+6y4+9=0
3) x7 +y*-4x-6y+9=0
Ary a4
Find the equation of the circle passing
through (-2, 14) and concentric with the
circle x? + y*—6x-4y-12=0.
1) x+y? -6x-4y-156=0
2) x74»? -6x+4y-156=0
3) x+y? -6x+4y+156=0
4) P+ y? +6x4+4y+156=0
19. Ifthe two circles x? + y?+2gr+c=Oand
x +y?-2fy-c=0 have equal radius
then locus of (g, f) is
I x+y =c? 2)x-y? =2¢
3) x-y? =e? 4) 2 4+y =2
The line x+y=I cuts the coordinate axes
at Pand Q and a line perpendicular to it
meet the axes in R and S. The equation to
the locus of the point of intersection of the
20.
aaa
lines PS and QR is
I xtty?=l
2) P+ y?-2x-3y=0
3) xP +y?—x-y=0
A +ytxty=0
POWER OF A POINT POSITION OFA
POINT WRT CIRCLE
21. Ifa line is drawn through a point A(3,4) te
cut the circle x? +? =4 at P andQ then
AP.AQ=
22. Geometric Mean of shortest and farthes
distances from a point w.nt a circle S=0 i
DS, DWSul 35% 4) Jor
23. The shortest distance from (-2, 14) to th:
circle x°+y?—6x—4y-12=0 is
24, The power of(1,1) with respect to th
circle x? + y?-4x43y+k=0i
3 then 'k' is
25. The equation of the circle with centr
(3,2) and the power of (1,-2) wart the circ!
x? +y?=1 as radius is
Ix? + -6x-4y-3=0
2)x° + y? -3x-2y-3=0
3)x° +? +6x+4y-3=0
4)x +y*-6x-4y43=0
26. A chord of length. 24, units is at
distance of 5 units from the center of
circle then its radius is
5 i a io
27. Ia chord of circle x? + y? = g makes equ
intercepts of length ‘a’ on the coordina
axes then al <
28. If the chord y=mx+1 of the cire
2° +y* =1 subtends an angle of mesau
45° at the major segment of the circle th:
m= 29. The least distance of the
30.
3.
aa
34,
35,
. Ifone diameter of the circle, given by the
POSITION OF LINE W.R.T CIRCLE
AND RELATED CONDITIONS
1 B8x-4y+73=0
from the circle
16x? +16y? + 48x—8y—43 = Ois
1)(x42)' +(y-l) =
2) (x+2)' +(y-1) =16
3) (x-2)' +(y4l) =4
4) (x-2)' +(y+l)’ =16
2)2v5 2
1) V5/2_ 2)2N5 3)3V5_ 4) 4V5_| 36, Theparametricequations x=”
The equation of the circle with centre at
(4,3) and touching the line 5x-12y-10=0 is
|. y= Lar represents a circle whose radius
2) x+y? +6x-8y+16=0 is
es a 22a 3) Ba 4) 4a
37. To the circle x+y? +8x—4y+4=0
4) x? +? -24x-10y +144 =0
Ifa straight line through c(-v8.8)
making an angle] 35° with the x-axis
and cuts the circle x =5cos0,y = Ssin0
in points A and B then AB=
ppation 67 45-2
chord of a circle S, whose centre is at
(-3,2) then the radius of S is
PARAMETRIC EQUATIONS OF
CIRCLE
Parametric equation of the circle
x+y? =16 are
1) x=4c0s0,y=4sin0
2) x=4cos0,y=4tand
3) x=4cosh0,y=4sinhO
4) x=4sec0,y =4tand
Locus of the point (sec AO, tanh 0) is
I xetty?=1 Qxt-y=l
xetyte]=0 4x -yPaxty
Circle on which the coordinates of any point
are (2+4cos0,-1+4sind), where 0 is
parameter. is 1
x
tangent at the point 9 = 4s
I)x+y+2-4V2 =0
2) x-y+2-4V2=0
3) xty+24+4V2=0
4) x-y-2-4V2 =0
‘TANGENT, NORMAL CONDITIONS.
AND PROPERTIES
38. The tangent to the circle
4y?—4x4+2y+h=0 at (11) is
x-2y+1=0 then k=
|. The equations of the tangents to the circle
x? + y? =25 with slope 2 is
1) y=2x4V5 2) y=2xt2V5
3) y=2xt3V5 4) p=2xe5V5
x is a tangent to a circle with centre
40. Ify:
(1,1) then the other tangent drawn through
(0,0) to the circle is
1)3y=x 2) y=-3x
3) y=2« 4) 3y=-2x
41. Ifthe lines 3x—4y +4 =0 and
6x By —7=0 ate tangents to a circle,
then the radius of the circle is 42. The length of the tangent from (1,1) to
the circle 2x7 +2)? +5x+3y+1=0 i
pvis72 2)3 3)2 41
43. Ifthe tangent at the point P on the circle
x+y? +6x+6y=2 meets the straight
line Sx-2y+6=0 at a point Q on the
y—axis then the length of PQ is
44, The lengths of the tangent drawn from any
point on the circle 15x’+15y’-48x+64y = 0
to the two circles 5x°+5y?-24x+32y+75 =
and 5x°+5y*48x+64y+300 = 0 are in the
ratio of
12 -2)23 334 YAS
45. The normal at(1,1) to the circle
xty'-4x+6y-4=0 is
1) 4x+3y=7 2) Axty=5
3) xty=2 4) Ax-y=5
46. If the tangent at P on the circle
x? +y? =a’ cuts two parallel tangents of
the circle at A and B then PA.PB =
a 2) @ 3)2a 4) 2a?
47. 'O' is the origin and A, (x, , y,) where k=
1,2 are two points. If the circles are
described on OA, and OA, as diameters,
then the length of their common chord is
equal to
1
D&y-SYy 2) Fy - Xl
al [ay Ni
3) FAA, 4) td,
48. Theline 4y—3x+4=0 touches the circle
x+y? —4x—-8y—5=0 then 1=
49. If the line y=x touches the circle
x+y?+2ex+2fy+e=0at P where
OP=6y7 then c=
50. The circle to which two tangents can be
drawn from origin is
1) xP +y?-8x-4y-3=0
2) x+y? +4x42y4+2=0
3) x+y? -8x+6y+1=0
4) both (2) &(3)
Si.
52.
53.
54,
55.
56.
ANGLE BETWEEN TANGENTS
‘The condition that the pair of tangents
drawn from origin to circle
x+y? +2gx+2fy+e=0 may beat
right angle is
1) gt free 2) gt fr =2c
3) gt+fi+2ce=0 4) g’-f?=2c
Slopes of tangents through(7, 1) to the
circle x? + y? =25 satisfy the equation
1) 12m? +7m+12=0
2) 12m? -7m+12=0
3) 12m? +7m—12=0
4) 12m? -7m—-12=0
Angle between tangents drawn from a point
P to circle x+y? -4x-8y +8 =0i860"
then length of chord of contact of P is
‘Tangents AB and AC are drawn to the circle
xt+y?-2x+4y+1=0 from A(0,1) then
equation of circle passing through A,B and
Cis
1) x+y txt y42=0
2) xP +y?-xty-2=0
3) xP +y?-x-y-2=0
4) xP +y?-x-y+2=0
Locus of the point of intersection of
tangents to the circle 7+)? +2x+4y-1=0
which include an angle of 69° is
1) x+y? +2x4+4y-19=0
2) x+y? +2x4+4y+19=0
3) x+y? -2x-4y-19=0
4) x°+y?-2x-4y419=0
The locus of the point of intersection of the
two tangents drawn to the circle
x? +? =a? which include an angle q is
? cosec* a/2
P cot? a/2 ee eee ec
57. The locus of the point of intersection of
perpendicular tangents drawn to each of
circles + y?=16 and + y*=9 isa
circle whose diameter is
The locus of the point of intersection of
the tangents to the circle x =cosd,
y=rsin@ at the points whose
parametric angles differ by a is
2
Dx ey =r Daxtayt oar?
33 +y*)=27? 4307 45°) S47?
. Locus of point of intersection of
perpendicular tangents to the circle
x+y'-4x-6y-1=0 is
—4x-6y-15=0
~4x-6y+15=0
3) ety? -4x-3y-15=0
4) P+? +4x46y-15=0
Tangents to x? + y’
inclinations @ and f intersect at P. If
cota +cot B =0 then the locus of P is
I)xty=0 2)x-y=0 3)xy=0 4) xy=a*
CIRCLES TOUCHING THE
COORDINATEAXES, INTERCEPTS ON
THEAXES
61. The circle with centre (4,-1) and touching
x-axis is
60. =a’ having
1) xP 4? -8x+2y416=
2) x+y? +18x-2y-16=0
3) P+ —4x+y+4=0
4) Vey tl4x-y+4=0
02, ich has both
ich passes
‘The equation of the circ
the axes as its tangents and w!
through the point (1,2)
I) x24 y- 2x4 2y-1=0
2)xt+y-2x+2y+1=0
3)x?+y?-2x-2y+1=0
4) +y-2x-2y-1=0
Equation of circles touching x-axis at the
origin and the line 4x-3y+24=0 are
63,
64,
65.
68.
69.
70.
. Th
1) P+)? -6y=0, x7 +7 +24y=0,
2) t4y?42y=0, x7 +y?-18y=0,
3) xt +y? + 18x =0, x7 +? -8x =0,
4) P4y?44x=0, x7 +37 -16x=0
Equations of circles which touch both the
axes and whose centres are at a distance
of 2/2 units from origin are
I) Pty t4rb4y+4=0
2) x+y’ £2xt2y+4=0
3) x+y txt yt4=0
4) x +y?-4=0
The equation of the cirlce in the first
aquadrant touching each coordinate axis
ata distance of one unit from the origin is
3) x+y? -2x-2y
+y°-2x4+2y-150
. They-intercept of the circle
vty? +4xt8y—5=0 is
laa wid 3)6 = 412
ercept made by the circle with
centre (2, 3) and radius 6 on y-axis is
Nisv2 2) 12¥2 3) 82 462
The centre of the circle passing through
origin and making intercepts 8 and~4 on x
If the circle x?4y?-2x¢4y+
complete one revolution on the positive
direction of x-axis then the centre in new
position is
(1,2
3) (1+ 4n,-2)
Ifthe line hx + ky
2) (1,4n-2)
4) (1-4n,-2)
/a touches the circle
x? +y? =a? then the locus of (hk) is circle of radius
1 1
1) - 2) a? 3)a 4) 2
71. 2x+y=0is the equation of a diameter of
the circle which touches the lines
4x-3y+10=0 and 4x-3y-30-0. The centre
and radius of the circle are
1) (-2,1);4 2) (1-2); 8
3)(1,-2);4 4) (1,-2);16
72. Ifa circle of radius 2 touches X-axis at
(1,0) then its centre may be
DAV 0,220
i 3) (-1, 2) CL, -2) 4) (-1, 2) (-1, -2)
73. Centre of the circle toucing y-axis at (0,3)
and making an intercept 2 units on positive
X-axis is
1 (vi0,3) 2) (3,v10)
3) (-V10,3) 4) (-Vi0,-3)
74. Radius of a circle which touch the both axes
x
and the line at 7 =1 being the centre lies
in first quadrant
ab ab
Gab sJarb ) arbsJarb
ab ®
Varbene Fp
75. The radius of the circle which touches y-
axis at (0,0) and passes through the point
(b,c) is
B+c?
oR
lal
7 Dare)
76. Centre of a circle wich touches both the
axes and the line 3x —4y+8=0 and lies
in the third quadrants
1) (-1-1) 2)(-2,-2)
3) (-3,-1) 4)(-3,-3)
77. Equation of circles which touch both the
axes and also the line x = k (k>0) is
pa AY
1 2352
1) Pty esha
: B
2) xy bbethy 0
2 KR
aay beets
4) ety tke-ky+—=0
78, The circle ax? + ay? + 2gx + 2fy +¢=0
meets the x-axis in two points on opposite
sides of the origin if
Iac<0 2)ac>03)a>0 4)c>0
79. If the circlex?+y?+2er+2f+e=0
touches x - axis at (x,, 0) then x, is the
repeated root of
1) x24 2gx+
3) xt—2gx-
80. The circle x?+y?-2ar—2ay+a
touches axes of co-ordinates at
1) a), (0,0) 2) (a,0), (0,0)
3) @0), (0a) 4) (0,2), (a)
CHORD OF CONTACT, POLE AND
POLAR
81. The chord of contact of (2,1) want to the
circle x+y? +4x4+4y41=0 is
1) 2x+y+7=0
3)3xt4y+1=0
82. Equation of tangent
2) 4x+3y+7=0
4) not existing
to the circle, at th
point (1,-1) , whose centre is the Point of
intersection of the straight lines x— y=]
and 2x+y=3is
N)3x-y-4=0 2) x4 4y43=0
3) x-3y-4=0 -4)4x4y—320
83. Pole of diameter ofa circle wr to the samt
circle lies
1) inside of the circle 2) outside of the circk
3) onthe circle 4) Does not exist
84. The polars of three Points w.r. to a giver
circle are concurrent then the three point!
are
1) collinear je
2) forman equilateral
AS oe R
85.
86.
87.
88.
89.
90.
1.
3) forma right angled triangle
4) passes through the centre of the circle
The polar of (2,-1)w.
vy t+ore4y is Sxty+h=0
then k=
pole of 3x+5y+17=0 wart the circle
ty +4xt+6y4+9=0 is
11,2) 2)(2) 3) (2) 4)(2,1)
Ifaxtby+e=0 is the polar of (1,1) wrt the
circle x? + y*—2x+2y+1=OandH.CF of
a,b,cis equal to one then q? +5? +? =
The length of chord of contact of the point
(3,6) with respect to the circle
reyes
ye 26N5 3) V5 5
A tangent at a point on the circle
x2 +y? =a” intersect a concentric circle
1S" at Pand Q. The tangents of this circle
at P, Q meet on the circle x? + * =5* then
the equation of concentric circle is
2) v+yh=ab
s
1) P4+y? =a
veya dts) PR AS
‘The pair of tangents from (2,1) to the circle
x+y =4is
1) 3x? + 4xy+16x+8y+20=0
2) 3x? +4xy+16x—By+20=0
3) 3x? +4xy__16x—8y+20=0
4) 3x -dxy-16x+8y-20=0
The pair of tangents from origin to the
circle
x+y +4x4+2y+3=0 is
1) (x+y) =3(x? +9")
2) (ax+2y)' =3(27 +9")
3) (2x-y) =3(x? +y*) 4) not existing
92.
93.
94,
95.
96.
97.
98.
CHORD WITH MID POINT
The equation of the chord of
x24 y? 4x4 6y +3 =O whose mid pointis
(1,-2) is
1) xtyH1=0 2) 2xt3y+4=0
3) x 4) not existing
The locus of middle points of the chords of
the circle x + y* =a” subtending an angle
"a! at the centre is
1) x+y? =acosa/2
2) x? +y? =a’ cosa/2
3) x? + y? =a’ cos’ @/2
4) x24 y? =cos*a/2
The locus of midpoints of the chord of the
circle x? +y? =25 which pass through a
fixed point (4,6) is a circle .The radius of
that circle is
nye D2 DVB AVi0
Locus of mid points of chords to the
circle x? + y? -8x+6y +20 =Owhich are
parallel to the line 3x+4y+5=0 is
1) 3xt4y-25-0 2) 4xt3y+5=0
3) 4x-3y-2: 4) 4x-3y+25=0
Locus of midpoints of chords of circle
x? +y? = /?having a constant length ‘27
is
2
x+y -P Qxetysr-P
3) t4ytaa? a) ateyt=P ar
Let 'C’ be the circle with centre (0,0) and
radius3 units, The equation of the locus of
the mid points of chords of the circle 'C’
2n
that subtend an angle of =~ at its eentre
is
3
yPryras Qxerrye
2,227 2pyad
= yee
3) Pt aT eras
Number of positions of P such that Where A= (1,2), B= (1,6)
(N.T)
99. If the tangent at (3 ,-4) to the circle
x +y'-4x+2y-5=0cuts the circle
x+y? +16x+2y+10=0in Aand B then
the midpoint of AB is
1) (6,-7) 2) 2, -1)
3) 2,1) 4) (6,4)
CONJUGATE POINTS, CONJUGATE
LINES
100.If 4, and 7, are the lengths of tangents
drawn from two conjugate points A, B then
242
ft+h=
IAB 2)2AB_ 3) 4p? 4) 2.48?
101. If (1,1),(k,2) are conjugate points with
respect to the circle
x+y? +8x+2y+3=0, then k=
1-12 2)-12/73)-12/5 4) -4
102. The points (3,2), (2,3) w.rt the circle
x+y? =12 are
1) extremities ofa diameter 2) conjugatepoints
3) Inverse points 4) lie on the circle
103. If 3x+2y=3 and 2x+5y=1 are conjugate
lines wat the circle x? + y? =? then ,2=
4 3
Dig Re
104, If the lines 2x+3y -4=0 and kx + 4y-2=
0 are conjugate with respect to the circle
x+y'=4 thenk-1=
105. For the circle x? + y?-2x-4y—4 =0,the
lines 2x+3y-1=0 ,2x+y+5=0 are
1) perpendicular tangents 2) conjugate
3) parallel tangents 4) perpendicular chords
INVERSE POINTS
106. The inverse point of (2 ,-3) w.rt to circle
16
az
x+y? +6x—4y-12=0 is
11
»(23)
1
aa (3-3)
107. If the inverse of P(-3,5) w.nt to a circle is
(1,3) then polar of P w.rt to the circle is
1) xt2y=7 2) 2x-2y+11=0
3) 2x-y#1=0 4) 2x-y-1=0
CIRCUMCIRCLE, INCIRCLE AND
CONCYCLIC POINTS
108. The centre of the circle circumscribing
the square whose three sides are
3x4 y=22,x-3y=14 and 3x4 y=62 ist
034)
3) (27,3)
2) (16,-6)
0(2
The minimum distance between the circle
x? +y? =9 andthe curve 2x+10y +6xy=1
is
109,
1
D2 22 N32 4 3-Fy
Ifa circle is inscribed ina square of
side 10, so that the circle touches the
four sides of the square internally then
radius of the circle is
110 -)sJ2_ 3104) {
. Centre of the circle inscribed in a rectangle
formed by the lines x?_gy412=0 and
y-14y+40=0 is
14,7) 2)(7,4) 3) (9,4) 4) (4,9)
- Ifthe points (0,0), (2,0) ,(0,-2), and (k,-2)
are concyclic then k=
If the points(2,0)(0, 1),4,0)and(0,a) are
concyclic then a=
114, An equilateral triangle is inscribed in the
circle having the radius r then its side is
DBr DF
acireleand P(x,, y,) isan external
110.
=
j
u
11
B
IL
oF
1)3r 4)2r
115. S=0is
point to it Pj and pp are tangents to S= |
0 from the point P, A and B are points of. contacts of tangents. The centre of circum
circle of A PAB is
Dxy+sn+S)
oft) o( BZ)
116, ABCD isa square with side ‘a’. If AB and
‘AD are taken as positive coordinate axes
then equation of circle circumscribing the
square is
1x? +y?-ax-ay=0
1) (u-8. 4-S)
f
2x ty +axtay=0
3) 7 +y?-ax+ay=0
4) xP +y? +ax-ay=0
117.Aright angled issosceles triangle is
inscribed in the circle
x +y?—4x-2y—4=0 then length of
its side is
p2 222 3)3/2 4) aD
118.A square is inscribed in the circle
x?+y?—2x48y-8=0 whose diagonals
are parallel to axes and a vertex in the
first quadrant isA then OA is
DI 2) V2 3)2V2 4)3
119. The circle passing through (t , 1) , (1, t)
and (t , t) for all values of t also passes
through
10,0) 2)(1,1) 3)(-1) 4) (¢1-1)
120. The triangle PQR is inscribed in the circle
x+y? =25.1fQ=(3,4) and R=(-4,3) then
ZOPR=
x a
3) Ai NG
x «
D2 soPlse
121. ABCD is square of unit area. A circle is
tangent to two sides of ABCD and passes
through exactly one of its vertices. The
radius of the circle is
1
N2-V2 2) y2-1 3) 5
1
Dr
RELATIVE POSITION OF TWO
CIRCLES
122. The circles x? + y?-12x+8y+48=0,
x+y? -4x+2y-4=0 are
1) intersecting
2) touching externally
3) touching internally
4) one is lying inside the other
123, The circles x? +? -2x-4y-20=0,
x+y? +4x-2y+4=0 are
1) one lies out side the other
2) one lies completely inside the other
3)touch externally 4) touch internally
124.The number of common tangents to
x+y? =256,(x-3)' +(y—4) =121 is
Ione 2)two 3) four 4) zero
125.The internal centre of similitude of the
circles x? +y?-2x+4y+4=0,
x+y?+4x-2y+1=0 divides the
segment joining their centres in the ratio
HE2 2)2:1 3)-1:24)-2:1
126.The external centre of similitude of the
circle x7 + y?-12x+8y+48=0 and
v+y-4x+2y-4=0 divides
segment joining centres in the ratio
12:3 2)3:2.3)-2:3 4)-3:2
127. The equation of the circle circumscribing
the triangle whose sides are the lines
yaxt2,3y=4x, 2y=3x is
the
1)x? + y?-8x-2y=0
x+y? -46x+22y=0
3) xP ty?—4x+l1y=0
4) P+ y?-T2x424y=0
128. If two circles touching both the axes are
passing through (2, 3) then length of their
common chord is
129.1f the
w+y?=dy-4y+4=0 have exactly
three real common tangents then 4 =
130.The
wey
circles y°+y?=2 and
circles y°+y?=ay and
two
*(¢> 0) touch each other if a — ———— ee e—
2)a=20
3) ¢ 4) 2jalec
| .
131. The centre of the circle passing through
the points (0, 0) (1,0) and touching the
circle x? +y? =9 is
(545)
TRANSVERSE AND DIRECT COMMON
TANGENTS
132. If the distance between the centres of two
circles of radii 3,4 is 25 then the length of
the transverse common tangent is
1
133.1f (3) is a centre of similitude for
the circles x? +)? =] and
x? +y?-2x—6y—6=0, then the length
of common tangent of the circles is
Dee
D3 3
31 4) Cannot be determined
134.The common tangents to the circles
x+y? -6x=0,x7 +9? +2x=0 form
1)Right angled triangle 2) Isosceles triangle
3) Equilateral triangle
4) Isosceles right angled triangle
AREAS FORMED BY CIRCLES
135. A rectangle ABCD is inscribed in a circle
with a diameter lying along the line
3y = x10. IfA=(-6, 7), B= (4,7) then
area of the rectangle in sq. units is
136. Let AB be the chord 4x-3y+5=0 of the circle
xt+y?-2x+4y—20=0.1fC=(7, 1) then
the area of triangle ABC is
1) 15 sq.uint 2) 20 sq.unit
3) 24 sq.unit 4) 45 sq.umit
PROBLEMS BASED ON LOCUS
137. A and B are two fixed points. The locus of
sinB
P such that in APAB, aan isa constant
(#1) is___ #F
Iacircle 2) pair of lines
3) part ofacircle 4) line parallel to BC
138.A circle of constant radius 3k passes
through (0,0) and cuts the axes in A and B
then the locus of centroid of triangle OAB
is
2) r+y=2e
3) 74+ y? =3h 4) ery sae
139.A rod PQ of length 2a slides with its
ends on the axes. The locus of the
circumcentre of AOPQ is
Nxty=k
Dxt+y=2a — Dat +y? = 4a?
—
4x2 +y’
3)x? + y? =3a?
140. Chords of the circle
x+y? + 2gx + 2fy + c= 0 subtends a right
angle at the origin. The locus of the feet
of the perpendiculars from the origin to
these chords is
1) ¢+y text fy+c=0
2) 2t+y)+extfy+e=0
Ax ty + ext fy) +o50
4) xt+yt+2 (ext fy+c)=0
The locus of the point from which the
length of the tangent to the circle
141.
x+y? -2x-4y4+4=0 is 3 units is
Ix? +y?-2x-4y-9=0
2)x* +y*-2x-4y—4=0
3)x° + y-2x-4y-3=0 |
4)x? + y? -2x-4y-5 20?
142.Locus of the point of intersection of
perpendicular tangents to the circle
x+y? =10 is
Dx ty’ 20
3) x?4y?=10 4) x2 Hy? =100
143. Locus of the point of intersection of
perpendicular tangents drawn one to cach |
ofthe circles x? + y® land x? 4y? =12
im
2 ety? Dxrtyrsd 2) xP 4y? =20
3)axt ty? = 208 (4) xP 457 =16
144, The locus of points from which lengths of
tangents to the | two circles
vay? +4x43=0 :
x+y? -6x+5= Oar
acircle with centre
1) 6,0) 2) (-6,0)
3) (0,6) 4) (0,-6)
145.A rod AB of length 4 units moves
horizontally with its left end A always on
the ratio 2 : 3 is
the circle x* + y?-4x-18y—29 =0 then
the locus of the other end Bis
1) x? +y?-12x-8y+3=0
2) x? +y?-12x-18y+3=0
3) Pty? +4x-8y-29=0
4) xP 4y?-4x-l6y +19=0
146. The locus of the centre of the circle
(xeosa tysin @ - a)? +(x sina -y Coser -
by =K if a varies, is
Dxtysa 2xrty=b
ayxttysatb 4x +yaah
147.The jocus of the foot of the
perpendicular drawn from origin toa
variable line passing through fixed point
(2,3) is a circle whose diameter is
vs 9B yas ovis
148. Two rods of lengths ‘a’and ‘b’slide along
coordinate axes such that their ends are
concyclic.Locus of the centre of the circle
is !
1) 40? +y?)=a° +0?
240? + y*)=a? -B*
3) (x? -y*) =a" -0°
4) xy =ab
149, The locus of the point (/, m). If the line
‘x+my=I toughes the cirele x? + y* =a@is
2) ax? +2y? =a"
I) x+y? =2a?
)a@(itsy yal ANa(e+y)=2
150. A variable circle passes through the fixed
point (2, 0) and touches y-axis then the
locus of its centre is
1) Cirele 2) parabola
3) Ellipse 4) stright line
151. Acircle passes through A (1,1) and touches
x -axis then the locus of the other end of
the diameter through 'A' is
1) (x41) =4y 2) (y-l)’ = 4x
3) (x-1=4y 4) (yl) = 4x
152.The tangent at any point to the
circle x? +y? =r? meets the coordinate
axes at Aand B.If the lines drawn parallel
to axes through A and B meet at P then
locus of P is
x
153. The locus of the point which divides the
join of A(-1, 1) and a variable point P on
the circle x? +? =4 in the ratio 3:2 is
1) 25(x* +y?)+20(x+y)+28=0
2) 25(x? + y*)-20(x+ y)+28=0
3) 25(x° + y*)+20(x-y)+28=0
4) 25(x? + y?)+20(x—y)-28=0
154. Locus of the centre of the circle which
touches x? + y*-6x-6y +14=0
externally and also y-axis is
1) y>-6x-10y-14=0
2) y?-6x+10y+14=0
3) y +6x+10y+14=0
4) y?-6y-10x+14=0
155.1f x? +y? =16,x’ +y° =36 are two
circles and P and Q move respectively on
these circles such that PQ=4 then the locus i ius is i i i ‘straight lines is cer
i i i ircle of radiusis | 2) Point of intersection of straig nthe
haley Giscentre and radius is AG
3)
» 20 2 V2 3) 30 4) 32 4) radiusis2a
KEY 5) r=CP *)?
DI 2)2 31 a4 6) (2,3)Let (h, k) be the centre of the circle. Then|
5)3 6)4 14 8)2 k=h-—1. Therefore, the equation of circle jg
9-16 10)2 11)1 12)3 ; : :
Dot 143400 15)2.00 16) 1 givenby (x— A) +Ly-(4-1)] =9...0)
17)2 18)1 19)2 20)3 given that the circle passes through the point
2121 —22)2——«-23)8.00 24) 2.00 mn
351 -26)3.--27)4.00_28)-l (7.3) and hence we get
292 30)3.- 31) 10-32) 8.66 (7-0) +[3-(n-]} =9
3) | 301 35)4 | 30)2 So
37) 1: +i «:38) 0,00 _39)4. +. 40)4 therefore, the required equations of the
CUO re 2300 et circles are x? + y" -8x-6y +16=0
45)246)2.-47)4_— 48) -35
49)72 50)4— 51)2 52)4
53)6.00 54)2 55)1 56)1 7) Drawadiagram
57)1058)4 —59)1_— 60). | 8) Drawadiagram
61)1 62)3 63) 1 64) 1 7 4
65) 1 66) 1
a 3 a 4 oH 3 a } 10) Point of intersection of normals is centre
7B)1 74)375)1 16) 2 NM) vey
71 -78)1_—«79)1~—«80)3._—| 12) G)ABwillbe diameter.
81)2 82)2 §3)4 84) 1
85)3.00 86)2 87) 5.00 88)1
89)2 90)3.-91)1-—-92)3—‘| 13) addingtwo equations
93)3 94)3 95)3 96) 2 14) formtriangle
974 98)0- 991 100)3. | 45) 05
1013 107)9 1031 one | ee |
105)2106)4 107)3 105) | 16) Image ofthe centre (3,2) withrespect tot
109)2-110)4—«M)1.—«112)2.00 Tine is 1, —x+y-1=0=1
113)8.00 114)2 115)3—116)1_| 17) same centre
1173 118)2119)2120)3.-'| 18) Let the required cirele
De wae is a x4)? -6x—4y4.h=0. and it is passit
125)1 126)3.-127)1_—«128)1.41 Cee eos ae
129) 6.00 130)4 131)2 132) 24 through (~2,14)
133)3 134)3.135)80—136)3
) ) ) ) 19) 1, =n, and (g,f)=(x,y)
137)1 138)4139)4 140) :
141)4 —142)2-143)2-——144)2__| 20) Since Ris orthocentre QR is 3rd altitude. TI
145)2:146)3.—-147) 1 148) 3 circle on P,Q as diameter
149)3 150)2 151)3.—-152)1_| 21) itispower ofpoint (3,4)
153)4 — 154)4_155)2
22) y(CP+r)(CP=r) = JS,
xt+y?-l4x-12y+76=0
x ab
Since, AOB = = Centre= (24)
)
HINTS 23) CP-p
1) The equation of the circle on ABas diametet'+| 24) s =3
K(equation of AB)=0 i : Le ip
|| 25) C=,2), r=s,,
Paar UTA SO. UCB. OA ) avr -d? =24 dss
2) rod
28) chord makes 90° at the centre (0,0) and
homogonise wot
29) perpendicular distance - radius
30) verify
31) find line and solve with circle
32) Let'r'be the radius ofccircle S
=r=V25+50 => 53
33) x=rcos®;y=rsind
34) eliminate 9
35). From given conditions, we have
x=2+c0s0=(x-2)=4c0s0
and y=-1+4sinO > y+1=4sing
squaring and adding, we get
(x-2)' +(y+1) =16
36) squaring and adding
37) Apply (x+g)Cos0 + (y+ f)SinO-r=0
38) 5,=0
39) y=metrvitn?
40) verify
41) Distance between given parallel lines gives the
diameter ofthe circle.
42) Divide with 2 and apply ./S,,
43) -» The given line meets y-axis at Q (0, 3)
Then PQ= JS,
[-- PQ= length of tangent ftom Q to the circle ]
44) Find lengths of tangents from (0,0)
45) Passing through the origin
y
48) r=d
49) length oftangents from origin
50) Verify 5,, >0
51) r=;
52) y=mxtrv1+m? and substitute (7,1)
tan? =,
53) use '-9 = Te
tan =
54) DT
tn
55) use "> = Ss,
oe
56) tan 5 “J
57) director circle concept
58) angle between tangents is 7 —9
59) Director Circle
60) m,+m,=0
61) verify
62) foot ofthe perpendicular
63) verify
64) verify
65) since the equation can be written as
(x1) +(y-1)' =1 which represents a
circle touching both the axes with its centre
(1,1) and radius one unit.
66) 2/f?-¢
67) r=6,d=2and Wr —d? =8V2
68) Verify g*- f? =128.
2s x
SE) PAPB=a?
' ne
PAP =a 69)
41) Let (x94) 4. (22092) |) aa
fae ' re
Find the circles 04,, 04, as diameters 71) Lines are two parallel tangents, Centre lies on
diametre
ee ee ee a a
72) verify
73) draw the diagram
74) r=d 90)
18) fie 91)
76) Let abe the radius of the circle, then 92)
(a,-a) willbe centre and pependicular | 93) pet P(x,,y,) be mid point of chord and
distance from the centre to the given line
gives the radius of the circle. cos
77) draw the diagram 2 a
78) Onx-axis y=Othen ax? 42¢r+0=0 -u(1). 94) draw diagram
The roots of (1) are in opposite in sign 95) S,=8S,, and slope = -3/4
79) x*+2gx+e=0 has equalroots are x,,x, | 96), =5,
80) verify “4
81) S,=0 ae
) Sy 91) cos> ==
82) Ceareofees i$ 3) 98) AB=2r=4, r=2
Equation of circle is 1
( J ( 2 4y Maximum area = >*2%2-= 2 but given area=5
Joe II 9
3 3 «:, There isno such point.
2 8 16 2, 1_1,16
SUIS TY Sty tot > 99) 5,=0 and §,=5,,
=>3x7 +3y?-8x-2y=0 100) t? +3 =S,, +8, = AB
Equation oftangent at (1,—1) is 101) 8, =
3x-3y—4(x41)-(y-1)=0 102) S,, =0;CP.CQ #r*
=>-x-4y-3=0 103) 7? (yl, + mm.) = n\n,
=> x+4y43=0 104) r? (I, + mym,) = nin,
83) The polar ofany point on the diameter w.r. to | 105) r3(ij, + mym,) = nym,
same circle does not exist
- nena 106) foot of the perpendicular from (2,3) to it
— polar
85) S,=0 107)CP perpendicular to the polar of P
_ 108)Extremities ofa diagnal ofa square are
86) =0 and are
)_ S17 0.and compar (12, -14), (20, 2) centre = (16, -6)
87) S,=0 and compare : :
2 : ;
Cer eed 109) * (2cos* 0+10sin? 6 +6sin 0cos0) =1
89) take p(x,,y,) be any point on x? +y? =a? =1
and Q(a,f)be on x?+y?=b? and the |
polar of (a, 8) wart x? +y? =r? touches a
110) xa? =(2r)’
111) split lines and center is passing through point of
intersection of midway lines
H2cirele is x°+y?-ax—by=0 where
2
a=
113) aja, = 4b,
114) C0s30°= —, aisa side of i
2r
115) Mid point of Gp
116) draw diragram & observe
"17 <2 +22 = (277
118) Draw figure
119)t=1
120) mm, =—1
121) Equation of circle is
x? 4y?-2rx—-2ry+r?
(0,1)D cal).
A(,0)
B(L,0)
=rn4r+2=0, 9 r=24+v2
Put (ll), -22-V2 (rl)
122) CG =H +
123) C,C, <|,-]
125) r, 27, internally
126) r; :, externally
127)Circle passes through O. A(6,8), B(4,6).
Take circle equation as
Vey? +2gx+2fp=0
(c=0, vit passes through(0,0))
substitute A & B and solve for g.f.
128)Draw the diagram
129) GC, =" +n
130)c,¢,
131)
132) fa? -(, +n)
133)it is internal centre of similitude
134)draw the diagram, t:
cequailateral
3, Triangle is
135)equation of another diametre is x=—1
(perpendicular bisector of AB) therefore centre
is (-1,3) and BC=8, area = (10)(8)=80
136)base is length of common part and height is
perpendicular distance from (7,1) to its chord
of contact,
ap PA sinB
2 PB sinA
138) 00,0), A(a,0), B(0,b) and G(@x,y)
and AB=6k
139) AOPQ isright angle triangle
140)Let p(xy,»1) be the mid point ofa chord. OP
is Perpendicular to the chord. Equation of the
chord is x + yyy =37 +92 +)
Homogeneous the citcle equation with the help
of (1). Then use sum of the coefficients of
x?and y? is equal to zero
141)Let point be(x,,y,) and JS, =3
142)Director Circle
143) x7 +? =a? +b? 144) Bu 2
Is 3
145) Ais (h+rcos@,k+rsin@) and
Bis(x,y) and eliminateg
146)Eliminate ‘a’ fromx cos a +ysin @=aand
xsin a -ycos a =b
147)radiusis x7 + y", since itis the midpoint
148)2Jg?-e=a, 2f-c=b where
(-g.—f) =(x,¥) and simplify by squaring
and subtracting
14931 =
150) Definition of Parabola.
151) other end is (x,,),) and apply g* =c
152)Length of perpendicular ftom (0,0) to the line
ay
ee
42751;
glee
153)P is (2cos0,2sin8)
154) consider standard form and
GG, =" +1 andalso f? =c
155) Using applonius theorem
"P? +CQ? =2(CR? + RO”)
4y
=16+36-2{cr'($) }
= 52=2(CR+4) > CR=V22
eae
L
EXERCISE -II
Let (x,+)=0 be the equation of a circle.
If £(0,4)=0 has equal roots £=2.2 and
£(k.0)=0 has roots
centre of the circle is
Ifthe chord joining the points (2.-1), (1,-2)
subtends aright angle at the centre of the
circle, then its centre and radius are
(2,2) isa point on the circle
4 and Q is another point on the circle
such thatare pq = + (circumference).
a
The coordinates of Q are
1) (2-2) 2) (v2.-v2)
3) (-V2.2) 4) (2V2)
Let x(x—a)+»(y-1)=0 bea circle If
two chords from (a, 1) bisected by X-axis
are drawn to the circle then the condition
is
Da>s Dar<8 Na>4 A ar<4
If f(s {(x).f(y) for all x and y, (1) =2
and a, = f(n),n