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Page 1
Solved Examples on Circle
JEE Mains
Q1.A circle of radius 5 units touches both the axes and lies in first quadrant. If the circle makes one
complete roll on ?? -axis along the positive direction of ?? -axis, then its equation in the new position
is
(a) ?? ?? +?? ?? +???? ???? -???? ?? +?????? ?? ?? =??
(b) ?? ?? +?? ?? +???? ???? +???? ?? +?????? ?? ?? =??
(c) ?? ?? +?? ?? -???? ???? -???? ?? +?????? ?? ?? =??
(d) None of these
Ans : (d) The ?? -coordinate of the new position of the circle is 5+ circumferrence of the first circle =
5+10?? The ?? -coordinate is 5 and the radius is also 5 .
Hence, the equation of the circle in the new position is (?? -5-10?? )
2
+(?? -5)
2
=(5)
2
??? 2
+25+100?? 2
-10?? +100?? -20???? +?? 2
+25-10?? =25
??? 2
+?? 2
-20???? -10?? -10?? +100?? 2
+100?? +25=0
Q2. The abscissae of ?? and ?? are the roots of the equation ?? ?? +?? ???? -?? ?? =?? and their
ordinates are the roots of the equation ?? ?? +?? ???? -?? ?? =?? . The equation of the circle with ???? as
diameter is
(a) ?? ?? +?? ?? +?? ???? +?? ???? -?? ?? -?? ?? =??
(b) ?? ?? +?? ?? +?? ???? +???? -?? ?? -?? ?? =??
(c) ?? ?? +?? ?? +?? ???? +?? ???? +?? ?? +?? ?? =??
(d) None of these
Ans: (a) Let ?? 1
,?? 2
and ?? 1
,?? 2
be roots of ?? 2
+2???? -?? 2
=0 and ?? 2
+2???? -?? 2
=0 respectively.
Then, ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
and ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
The equation of the circle with ?? (?? 1
,?? 1
) and ?? (?? 2
,?? 2
) as the end points of diameter is
(?? -?? 1
)(?? -?? 2
)+(?? -?? 1
)(?? -?? 2
)=0
?? 2
+?? 2
-?? (?? 1
+?? 2
)-?? (?? 1
+?? 2
)+?? 1
?? 2
+?? 1
?? 2
=0; ?? 2
+?? 2
+2???? +2???? -?? 2
-?? 2
=0
Q3. Let ?? (?? ,?? )=?? be the equation of a circle. If ?? (?? ,?? )=?? has equal roots ?? =?? ,?? and
?? (?? ,?? )=?? has roots ?? =
?? ?? ,?? , then the centre of the circle is
(a) (?? ,
????
????
)
(b) (
????
????
,?? )
(c) (-?? ,
????
????
)
(d) None of these
Page 2
Solved Examples on Circle
JEE Mains
Q1.A circle of radius 5 units touches both the axes and lies in first quadrant. If the circle makes one
complete roll on ?? -axis along the positive direction of ?? -axis, then its equation in the new position
is
(a) ?? ?? +?? ?? +???? ???? -???? ?? +?????? ?? ?? =??
(b) ?? ?? +?? ?? +???? ???? +???? ?? +?????? ?? ?? =??
(c) ?? ?? +?? ?? -???? ???? -???? ?? +?????? ?? ?? =??
(d) None of these
Ans : (d) The ?? -coordinate of the new position of the circle is 5+ circumferrence of the first circle =
5+10?? The ?? -coordinate is 5 and the radius is also 5 .
Hence, the equation of the circle in the new position is (?? -5-10?? )
2
+(?? -5)
2
=(5)
2
??? 2
+25+100?? 2
-10?? +100?? -20???? +?? 2
+25-10?? =25
??? 2
+?? 2
-20???? -10?? -10?? +100?? 2
+100?? +25=0
Q2. The abscissae of ?? and ?? are the roots of the equation ?? ?? +?? ???? -?? ?? =?? and their
ordinates are the roots of the equation ?? ?? +?? ???? -?? ?? =?? . The equation of the circle with ???? as
diameter is
(a) ?? ?? +?? ?? +?? ???? +?? ???? -?? ?? -?? ?? =??
(b) ?? ?? +?? ?? +?? ???? +???? -?? ?? -?? ?? =??
(c) ?? ?? +?? ?? +?? ???? +?? ???? +?? ?? +?? ?? =??
(d) None of these
Ans: (a) Let ?? 1
,?? 2
and ?? 1
,?? 2
be roots of ?? 2
+2???? -?? 2
=0 and ?? 2
+2???? -?? 2
=0 respectively.
Then, ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
and ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
The equation of the circle with ?? (?? 1
,?? 1
) and ?? (?? 2
,?? 2
) as the end points of diameter is
(?? -?? 1
)(?? -?? 2
)+(?? -?? 1
)(?? -?? 2
)=0
?? 2
+?? 2
-?? (?? 1
+?? 2
)-?? (?? 1
+?? 2
)+?? 1
?? 2
+?? 1
?? 2
=0; ?? 2
+?? 2
+2???? +2???? -?? 2
-?? 2
=0
Q3. Let ?? (?? ,?? )=?? be the equation of a circle. If ?? (?? ,?? )=?? has equal roots ?? =?? ,?? and
?? (?? ,?? )=?? has roots ?? =
?? ?? ,?? , then the centre of the circle is
(a) (?? ,
????
????
)
(b) (
????
????
,?? )
(c) (-?? ,
????
????
)
(d) None of these
Ans : (b) ??? (?? ,?? )=?? 2
+?? 2
+2???? +2???? +?? =0
Now, ?? (0,?? )=?? 2
+2???? +?? =0 and its roots are 2, 2. ?2+2=-2?? ,2×2=?? , i.e. ?? =-2,?? =
4
?? (?? ,0)=?? 2
+2???? +?? =0, and its roots are
4
5
,5.
?
4
5
+5=-2?? ,
4
5
×5=?? , i.e., ?? =
-29
10
,?? =4. Hence, centre of the circle =(-?? ,-?? )=(
29
10
,2).
Q4. The range of values of a for which the point (?? ,?? ) is outside the circles ?? ?? +?? ?? +???? ?? =??
and ?? ?? +?? ?? -???? ?? +???? =?? is
(a) (-8,-?? )?(-?? ,?? )?(?? ,+8)
(b) (-?? ,-?? )
(c) (-8,-?? )?(-?? ,+8)
(d) None of these
Ans : (a) For circle, ?? 2
+?? 2
+10?? =0;
?? 2
+(4)
2
+10?? >0??? 2
+10?? +16>0?(?? +8)(?? +2)>0??? <-8 or ?? >-2
For circle, ?? 2
+?? 2
-12?? +20=0;?? 2
+(4)
2
-12?? +20>0??? 2
-12?? +36>0
?(?? -6)
2
>0??? ??? ~{6}
Taking common values from (i) and (ii), ?? ?(-8,-8)?(-2,6)?(6,+8).
Q5. If the straight line ?? =???? is outside the circle ?? ?? +?? ?? -???? ?? +???? =?? , then
(a) ?? >??
(b) ?? <??
(c) |?? |>??
(d) |?? |<??
Ans: (d) If the straight line ?? =???? is outside the given circle then perpendicular distance of line
from centre of circle > radius of circle
10
v1+?? 2
>v10 ? (1+?? 2
)<10??? 2
<9 ?|?? |<3
Q6. The equations of any tangents to the circle ?? ?? +?? ?? -?? ?? +?? ?? -?? =?? is
(a) ?? =?? (?? -?? )+?? v?? +?? ?? -??
(b) ?? =???? +?? v?? +?? ??
(c) ?? =???? +?? v?? +?? ?? -??
(d) None of these
Ans : (a) Equation of circle is (?? -1)
2
+(?? +2)
2
=3
2
.
As any tangent to ?? 2
+?? 2
=3
2
is given by ?? =???? +3v1+?? 2
Any tangent to the given circle will be ?? +2=?? (?? -1)+3v1+?? 2
??? =?? (?? -1)+
3v1+?? 2
-2
Page 3
Solved Examples on Circle
JEE Mains
Q1.A circle of radius 5 units touches both the axes and lies in first quadrant. If the circle makes one
complete roll on ?? -axis along the positive direction of ?? -axis, then its equation in the new position
is
(a) ?? ?? +?? ?? +???? ???? -???? ?? +?????? ?? ?? =??
(b) ?? ?? +?? ?? +???? ???? +???? ?? +?????? ?? ?? =??
(c) ?? ?? +?? ?? -???? ???? -???? ?? +?????? ?? ?? =??
(d) None of these
Ans : (d) The ?? -coordinate of the new position of the circle is 5+ circumferrence of the first circle =
5+10?? The ?? -coordinate is 5 and the radius is also 5 .
Hence, the equation of the circle in the new position is (?? -5-10?? )
2
+(?? -5)
2
=(5)
2
??? 2
+25+100?? 2
-10?? +100?? -20???? +?? 2
+25-10?? =25
??? 2
+?? 2
-20???? -10?? -10?? +100?? 2
+100?? +25=0
Q2. The abscissae of ?? and ?? are the roots of the equation ?? ?? +?? ???? -?? ?? =?? and their
ordinates are the roots of the equation ?? ?? +?? ???? -?? ?? =?? . The equation of the circle with ???? as
diameter is
(a) ?? ?? +?? ?? +?? ???? +?? ???? -?? ?? -?? ?? =??
(b) ?? ?? +?? ?? +?? ???? +???? -?? ?? -?? ?? =??
(c) ?? ?? +?? ?? +?? ???? +?? ???? +?? ?? +?? ?? =??
(d) None of these
Ans: (a) Let ?? 1
,?? 2
and ?? 1
,?? 2
be roots of ?? 2
+2???? -?? 2
=0 and ?? 2
+2???? -?? 2
=0 respectively.
Then, ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
and ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
The equation of the circle with ?? (?? 1
,?? 1
) and ?? (?? 2
,?? 2
) as the end points of diameter is
(?? -?? 1
)(?? -?? 2
)+(?? -?? 1
)(?? -?? 2
)=0
?? 2
+?? 2
-?? (?? 1
+?? 2
)-?? (?? 1
+?? 2
)+?? 1
?? 2
+?? 1
?? 2
=0; ?? 2
+?? 2
+2???? +2???? -?? 2
-?? 2
=0
Q3. Let ?? (?? ,?? )=?? be the equation of a circle. If ?? (?? ,?? )=?? has equal roots ?? =?? ,?? and
?? (?? ,?? )=?? has roots ?? =
?? ?? ,?? , then the centre of the circle is
(a) (?? ,
????
????
)
(b) (
????
????
,?? )
(c) (-?? ,
????
????
)
(d) None of these
Ans : (b) ??? (?? ,?? )=?? 2
+?? 2
+2???? +2???? +?? =0
Now, ?? (0,?? )=?? 2
+2???? +?? =0 and its roots are 2, 2. ?2+2=-2?? ,2×2=?? , i.e. ?? =-2,?? =
4
?? (?? ,0)=?? 2
+2???? +?? =0, and its roots are
4
5
,5.
?
4
5
+5=-2?? ,
4
5
×5=?? , i.e., ?? =
-29
10
,?? =4. Hence, centre of the circle =(-?? ,-?? )=(
29
10
,2).
Q4. The range of values of a for which the point (?? ,?? ) is outside the circles ?? ?? +?? ?? +???? ?? =??
and ?? ?? +?? ?? -???? ?? +???? =?? is
(a) (-8,-?? )?(-?? ,?? )?(?? ,+8)
(b) (-?? ,-?? )
(c) (-8,-?? )?(-?? ,+8)
(d) None of these
Ans : (a) For circle, ?? 2
+?? 2
+10?? =0;
?? 2
+(4)
2
+10?? >0??? 2
+10?? +16>0?(?? +8)(?? +2)>0??? <-8 or ?? >-2
For circle, ?? 2
+?? 2
-12?? +20=0;?? 2
+(4)
2
-12?? +20>0??? 2
-12?? +36>0
?(?? -6)
2
>0??? ??? ~{6}
Taking common values from (i) and (ii), ?? ?(-8,-8)?(-2,6)?(6,+8).
Q5. If the straight line ?? =???? is outside the circle ?? ?? +?? ?? -???? ?? +???? =?? , then
(a) ?? >??
(b) ?? <??
(c) |?? |>??
(d) |?? |<??
Ans: (d) If the straight line ?? =???? is outside the given circle then perpendicular distance of line
from centre of circle > radius of circle
10
v1+?? 2
>v10 ? (1+?? 2
)<10??? 2
<9 ?|?? |<3
Q6. The equations of any tangents to the circle ?? ?? +?? ?? -?? ?? +?? ?? -?? =?? is
(a) ?? =?? (?? -?? )+?? v?? +?? ?? -??
(b) ?? =???? +?? v?? +?? ??
(c) ?? =???? +?? v?? +?? ?? -??
(d) None of these
Ans : (a) Equation of circle is (?? -1)
2
+(?? +2)
2
=3
2
.
As any tangent to ?? 2
+?? 2
=3
2
is given by ?? =???? +3v1+?? 2
Any tangent to the given circle will be ?? +2=?? (?? -1)+3v1+?? 2
??? =?? (?? -1)+
3v1+?? 2
-2
Q7. If the distances from the origin to the centres of three circles ?? ?? +?? ?? +?? ?? ?? ?? -?? ?? =?? (?? =
?? ,?? ,?? ) are in G.P. then the lengths of the tangents drawn to them from any point on the circle
?? ?? +?? ?? =?? ?? are in
(a) A.P.
(b) G.P .
(c) H.P.
(d) None of these
Ans : (b) The centres of the given circles are (-?? ?? ,0)(?? =1,2,3)
The distances from the origin to the centres are ?? ?? (?? =1,2,3). It is given that ?? 2
2
=?? 1
?? 3
.
Let ?? (h,?? ) be any point on the circle ?? 2
+?? 2
=?? 2
, then,
h
2
+?? 2
=?? 2
Now, ?? ?? = length of the tangent from (h,?? ) to ?? 2
+?? 2
+2?? ?? ?? -?? 2
=0
=vh
2
+?? 2
+2?? ?? h-?? 2
=v?? 2
+2?? ?? h-?? 2
=v2?? ?? h
[?h
2
+?? 2
=?? 2
and ?? =1,2,3]
Therefore, ?? 2
2
=2?? 2
h=2h(v?? 1
?? 3
)
[??? 2
2
=?? 1
?? 3
]
=v2h?? 1
v2h?? 3
=?? 1
?? 3
. Hence, ?? 1
,?? 2
,?? 3
are in G.P .
Q8. The angle between a pair of tangents drawn from a point ?? to the circle ?? ?? +?? ?? +?? ?? -?? ?? +
?? ?????? ?? ?? +???? ?????? ?? ?? =?? is
(a) ?? ?? +?? ?? +?? ?? -?? ?? +?? =??
(b) ?? ?? +?? ?? +?? ?? -?? ?? -?? =??
(c) ?? ?? +?? ?? +?? ?? -?? ?? -?? =??
(d) ?? ?? +?? ?? +?? ?? -?? ?? +?? =??
Ans : (d) The centre of the circle ?? 2
+?? 2
+4?? -6?? +9sin
2
?? +13cos
2
?? =0 is ?? (-2,3) and its
radius is
v2
2
+(-3)
2
-9sin
2
?? -13cos
2
?? =v4+9-9sin
2
?? -13cos
2
?? =2sin ??
Let ?? (h,?? ) be any point on the locus. The ??????? =?? . Also ??????? =?? /2 i.e. triangle ?????? is a right
angle triangle.
Thus sin ?? =
????
????
=
2sin ?? v(h+2)
2
+(?? -3)
2
?v(h+2)
2
+(?? -3)
2
=2?(h+2)
2
+(?? -3)
2
=4or h
2
+?? 2
+4h-6?? +9=0
Thus the required equation of the locus is ?? 2
+?? 2
+4?? -6?? +9=0.
Page 4
Solved Examples on Circle
JEE Mains
Q1.A circle of radius 5 units touches both the axes and lies in first quadrant. If the circle makes one
complete roll on ?? -axis along the positive direction of ?? -axis, then its equation in the new position
is
(a) ?? ?? +?? ?? +???? ???? -???? ?? +?????? ?? ?? =??
(b) ?? ?? +?? ?? +???? ???? +???? ?? +?????? ?? ?? =??
(c) ?? ?? +?? ?? -???? ???? -???? ?? +?????? ?? ?? =??
(d) None of these
Ans : (d) The ?? -coordinate of the new position of the circle is 5+ circumferrence of the first circle =
5+10?? The ?? -coordinate is 5 and the radius is also 5 .
Hence, the equation of the circle in the new position is (?? -5-10?? )
2
+(?? -5)
2
=(5)
2
??? 2
+25+100?? 2
-10?? +100?? -20???? +?? 2
+25-10?? =25
??? 2
+?? 2
-20???? -10?? -10?? +100?? 2
+100?? +25=0
Q2. The abscissae of ?? and ?? are the roots of the equation ?? ?? +?? ???? -?? ?? =?? and their
ordinates are the roots of the equation ?? ?? +?? ???? -?? ?? =?? . The equation of the circle with ???? as
diameter is
(a) ?? ?? +?? ?? +?? ???? +?? ???? -?? ?? -?? ?? =??
(b) ?? ?? +?? ?? +?? ???? +???? -?? ?? -?? ?? =??
(c) ?? ?? +?? ?? +?? ???? +?? ???? +?? ?? +?? ?? =??
(d) None of these
Ans: (a) Let ?? 1
,?? 2
and ?? 1
,?? 2
be roots of ?? 2
+2???? -?? 2
=0 and ?? 2
+2???? -?? 2
=0 respectively.
Then, ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
and ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
The equation of the circle with ?? (?? 1
,?? 1
) and ?? (?? 2
,?? 2
) as the end points of diameter is
(?? -?? 1
)(?? -?? 2
)+(?? -?? 1
)(?? -?? 2
)=0
?? 2
+?? 2
-?? (?? 1
+?? 2
)-?? (?? 1
+?? 2
)+?? 1
?? 2
+?? 1
?? 2
=0; ?? 2
+?? 2
+2???? +2???? -?? 2
-?? 2
=0
Q3. Let ?? (?? ,?? )=?? be the equation of a circle. If ?? (?? ,?? )=?? has equal roots ?? =?? ,?? and
?? (?? ,?? )=?? has roots ?? =
?? ?? ,?? , then the centre of the circle is
(a) (?? ,
????
????
)
(b) (
????
????
,?? )
(c) (-?? ,
????
????
)
(d) None of these
Ans : (b) ??? (?? ,?? )=?? 2
+?? 2
+2???? +2???? +?? =0
Now, ?? (0,?? )=?? 2
+2???? +?? =0 and its roots are 2, 2. ?2+2=-2?? ,2×2=?? , i.e. ?? =-2,?? =
4
?? (?? ,0)=?? 2
+2???? +?? =0, and its roots are
4
5
,5.
?
4
5
+5=-2?? ,
4
5
×5=?? , i.e., ?? =
-29
10
,?? =4. Hence, centre of the circle =(-?? ,-?? )=(
29
10
,2).
Q4. The range of values of a for which the point (?? ,?? ) is outside the circles ?? ?? +?? ?? +???? ?? =??
and ?? ?? +?? ?? -???? ?? +???? =?? is
(a) (-8,-?? )?(-?? ,?? )?(?? ,+8)
(b) (-?? ,-?? )
(c) (-8,-?? )?(-?? ,+8)
(d) None of these
Ans : (a) For circle, ?? 2
+?? 2
+10?? =0;
?? 2
+(4)
2
+10?? >0??? 2
+10?? +16>0?(?? +8)(?? +2)>0??? <-8 or ?? >-2
For circle, ?? 2
+?? 2
-12?? +20=0;?? 2
+(4)
2
-12?? +20>0??? 2
-12?? +36>0
?(?? -6)
2
>0??? ??? ~{6}
Taking common values from (i) and (ii), ?? ?(-8,-8)?(-2,6)?(6,+8).
Q5. If the straight line ?? =???? is outside the circle ?? ?? +?? ?? -???? ?? +???? =?? , then
(a) ?? >??
(b) ?? <??
(c) |?? |>??
(d) |?? |<??
Ans: (d) If the straight line ?? =???? is outside the given circle then perpendicular distance of line
from centre of circle > radius of circle
10
v1+?? 2
>v10 ? (1+?? 2
)<10??? 2
<9 ?|?? |<3
Q6. The equations of any tangents to the circle ?? ?? +?? ?? -?? ?? +?? ?? -?? =?? is
(a) ?? =?? (?? -?? )+?? v?? +?? ?? -??
(b) ?? =???? +?? v?? +?? ??
(c) ?? =???? +?? v?? +?? ?? -??
(d) None of these
Ans : (a) Equation of circle is (?? -1)
2
+(?? +2)
2
=3
2
.
As any tangent to ?? 2
+?? 2
=3
2
is given by ?? =???? +3v1+?? 2
Any tangent to the given circle will be ?? +2=?? (?? -1)+3v1+?? 2
??? =?? (?? -1)+
3v1+?? 2
-2
Q7. If the distances from the origin to the centres of three circles ?? ?? +?? ?? +?? ?? ?? ?? -?? ?? =?? (?? =
?? ,?? ,?? ) are in G.P. then the lengths of the tangents drawn to them from any point on the circle
?? ?? +?? ?? =?? ?? are in
(a) A.P.
(b) G.P .
(c) H.P.
(d) None of these
Ans : (b) The centres of the given circles are (-?? ?? ,0)(?? =1,2,3)
The distances from the origin to the centres are ?? ?? (?? =1,2,3). It is given that ?? 2
2
=?? 1
?? 3
.
Let ?? (h,?? ) be any point on the circle ?? 2
+?? 2
=?? 2
, then,
h
2
+?? 2
=?? 2
Now, ?? ?? = length of the tangent from (h,?? ) to ?? 2
+?? 2
+2?? ?? ?? -?? 2
=0
=vh
2
+?? 2
+2?? ?? h-?? 2
=v?? 2
+2?? ?? h-?? 2
=v2?? ?? h
[?h
2
+?? 2
=?? 2
and ?? =1,2,3]
Therefore, ?? 2
2
=2?? 2
h=2h(v?? 1
?? 3
)
[??? 2
2
=?? 1
?? 3
]
=v2h?? 1
v2h?? 3
=?? 1
?? 3
. Hence, ?? 1
,?? 2
,?? 3
are in G.P .
Q8. The angle between a pair of tangents drawn from a point ?? to the circle ?? ?? +?? ?? +?? ?? -?? ?? +
?? ?????? ?? ?? +???? ?????? ?? ?? =?? is
(a) ?? ?? +?? ?? +?? ?? -?? ?? +?? =??
(b) ?? ?? +?? ?? +?? ?? -?? ?? -?? =??
(c) ?? ?? +?? ?? +?? ?? -?? ?? -?? =??
(d) ?? ?? +?? ?? +?? ?? -?? ?? +?? =??
Ans : (d) The centre of the circle ?? 2
+?? 2
+4?? -6?? +9sin
2
?? +13cos
2
?? =0 is ?? (-2,3) and its
radius is
v2
2
+(-3)
2
-9sin
2
?? -13cos
2
?? =v4+9-9sin
2
?? -13cos
2
?? =2sin ??
Let ?? (h,?? ) be any point on the locus. The ??????? =?? . Also ??????? =?? /2 i.e. triangle ?????? is a right
angle triangle.
Thus sin ?? =
????
????
=
2sin ?? v(h+2)
2
+(?? -3)
2
?v(h+2)
2
+(?? -3)
2
=2?(h+2)
2
+(?? -3)
2
=4or h
2
+?? 2
+4h-6?? +9=0
Thus the required equation of the locus is ?? 2
+?? 2
+4?? -6?? +9=0.
Q9. Tangents are drawn from any point on the circle ?? ?? +?? ?? =?? ?? to the circle ?? ?? +?? ?? =?? ?? . If
the chord of contact touches the circle ?? ?? +?? ?? =?? ?? ,?? >?? , then
(a) a, b, c are in A.P .
(b) a, b, c are in G.P .
(c) a, b, c are in H.P . (d) a, c, b are in G.P .
Ans : (b) Chord of contact of any point (?? cos ?? ,?? sin ?? ) on 1
st
circle with respect to 2
nd
circle is
???? cos ?? +???? sin ?? =?? 2
This chord touches the circle ?? 2
+?? 2
=?? 2
,
Hence, Radius = Perpendicular distance of chord from centre.
?? =
?? 2
?? vcos
2
?? +sin
2
?? ??? 2
=???? .Hence ?? ,?? ,?? are in G.P .
Q10. A foot of the normal from the point (?? ,?? ) to a circle is (?? ,?? ) and a diameter of the circle has
the equation ?? ?? -?? =?? . Then the equation of the circle is
(a) ?? ?? +?? ?? +?? ?? -?? =??
(b) ?? ?? +?? ?? -?? ?? -?? =??
(c) ?? ?? +?? ?? -?? ?? -?? =??
(d) None of these.
Ans : (b) The line joining (4,3) and (2,1) is also along a diameter. So, the centre is the intersection of
the diameters 2?? -?? =2 and ?? -3=(?? -4). Solving these, the centre =(1,0)
? Radius = Distance between (1,0) and (2,1)=v2.
? Equation of circle (?? -1)
2
+?? 2
=(v2)
2
??? 2
+?? 2
-2?? -1=0
Q11. The polar of the point (?? ,-
?? ?? ) with respect to circle (?? -?? )
?? +?? ?? =?? is
(a) ?? ?? -???? ?? +?? =??
(b) ?? ?? -?? -???? =??
(c) ???? ?? -?? -???? =??
(d) ?? -???? ?? -?? =??
Ans : (b)
The polar of the point (5,-
1
2
) is ?? ?? 1
+?? ?? 1
+?? (?? +?? 1
)+?? (?? +?? 1
)+?? =0
?5?? -
1
2
?? -2(?? +5)+0+0=0?3?? -
?? 2
-10=0?6?? -?? -20=0.
Q12. The equation of the circle which touches the circle ?? ?? +?? ?? -?? ?? +?? ?? +???? =?? externally
and to which the lines ?? ?? -?? ???? -?? ?? +?? ?? =?? are normals, is
(a) ?? ?? +?? ?? -?? ?? -?? ?? -?? =??
(b) ?? ?? +?? ?? +?? ?? +?? ?? +?? =??
(c) ?? ?? +?? ?? -?? ?? -?? ?? +?? =??
(d) ?? ?? +?? ?? -?? ?? -?? ?? +?? =??
Page 5
Solved Examples on Circle
JEE Mains
Q1.A circle of radius 5 units touches both the axes and lies in first quadrant. If the circle makes one
complete roll on ?? -axis along the positive direction of ?? -axis, then its equation in the new position
is
(a) ?? ?? +?? ?? +???? ???? -???? ?? +?????? ?? ?? =??
(b) ?? ?? +?? ?? +???? ???? +???? ?? +?????? ?? ?? =??
(c) ?? ?? +?? ?? -???? ???? -???? ?? +?????? ?? ?? =??
(d) None of these
Ans : (d) The ?? -coordinate of the new position of the circle is 5+ circumferrence of the first circle =
5+10?? The ?? -coordinate is 5 and the radius is also 5 .
Hence, the equation of the circle in the new position is (?? -5-10?? )
2
+(?? -5)
2
=(5)
2
??? 2
+25+100?? 2
-10?? +100?? -20???? +?? 2
+25-10?? =25
??? 2
+?? 2
-20???? -10?? -10?? +100?? 2
+100?? +25=0
Q2. The abscissae of ?? and ?? are the roots of the equation ?? ?? +?? ???? -?? ?? =?? and their
ordinates are the roots of the equation ?? ?? +?? ???? -?? ?? =?? . The equation of the circle with ???? as
diameter is
(a) ?? ?? +?? ?? +?? ???? +?? ???? -?? ?? -?? ?? =??
(b) ?? ?? +?? ?? +?? ???? +???? -?? ?? -?? ?? =??
(c) ?? ?? +?? ?? +?? ???? +?? ???? +?? ?? +?? ?? =??
(d) None of these
Ans: (a) Let ?? 1
,?? 2
and ?? 1
,?? 2
be roots of ?? 2
+2???? -?? 2
=0 and ?? 2
+2???? -?? 2
=0 respectively.
Then, ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
and ?? 1
+?? 2
=-2?? ,?? 1
?? 2
=-?? 2
The equation of the circle with ?? (?? 1
,?? 1
) and ?? (?? 2
,?? 2
) as the end points of diameter is
(?? -?? 1
)(?? -?? 2
)+(?? -?? 1
)(?? -?? 2
)=0
?? 2
+?? 2
-?? (?? 1
+?? 2
)-?? (?? 1
+?? 2
)+?? 1
?? 2
+?? 1
?? 2
=0; ?? 2
+?? 2
+2???? +2???? -?? 2
-?? 2
=0
Q3. Let ?? (?? ,?? )=?? be the equation of a circle. If ?? (?? ,?? )=?? has equal roots ?? =?? ,?? and
?? (?? ,?? )=?? has roots ?? =
?? ?? ,?? , then the centre of the circle is
(a) (?? ,
????
????
)
(b) (
????
????
,?? )
(c) (-?? ,
????
????
)
(d) None of these
Ans : (b) ??? (?? ,?? )=?? 2
+?? 2
+2???? +2???? +?? =0
Now, ?? (0,?? )=?? 2
+2???? +?? =0 and its roots are 2, 2. ?2+2=-2?? ,2×2=?? , i.e. ?? =-2,?? =
4
?? (?? ,0)=?? 2
+2???? +?? =0, and its roots are
4
5
,5.
?
4
5
+5=-2?? ,
4
5
×5=?? , i.e., ?? =
-29
10
,?? =4. Hence, centre of the circle =(-?? ,-?? )=(
29
10
,2).
Q4. The range of values of a for which the point (?? ,?? ) is outside the circles ?? ?? +?? ?? +???? ?? =??
and ?? ?? +?? ?? -???? ?? +???? =?? is
(a) (-8,-?? )?(-?? ,?? )?(?? ,+8)
(b) (-?? ,-?? )
(c) (-8,-?? )?(-?? ,+8)
(d) None of these
Ans : (a) For circle, ?? 2
+?? 2
+10?? =0;
?? 2
+(4)
2
+10?? >0??? 2
+10?? +16>0?(?? +8)(?? +2)>0??? <-8 or ?? >-2
For circle, ?? 2
+?? 2
-12?? +20=0;?? 2
+(4)
2
-12?? +20>0??? 2
-12?? +36>0
?(?? -6)
2
>0??? ??? ~{6}
Taking common values from (i) and (ii), ?? ?(-8,-8)?(-2,6)?(6,+8).
Q5. If the straight line ?? =???? is outside the circle ?? ?? +?? ?? -???? ?? +???? =?? , then
(a) ?? >??
(b) ?? <??
(c) |?? |>??
(d) |?? |<??
Ans: (d) If the straight line ?? =???? is outside the given circle then perpendicular distance of line
from centre of circle > radius of circle
10
v1+?? 2
>v10 ? (1+?? 2
)<10??? 2
<9 ?|?? |<3
Q6. The equations of any tangents to the circle ?? ?? +?? ?? -?? ?? +?? ?? -?? =?? is
(a) ?? =?? (?? -?? )+?? v?? +?? ?? -??
(b) ?? =???? +?? v?? +?? ??
(c) ?? =???? +?? v?? +?? ?? -??
(d) None of these
Ans : (a) Equation of circle is (?? -1)
2
+(?? +2)
2
=3
2
.
As any tangent to ?? 2
+?? 2
=3
2
is given by ?? =???? +3v1+?? 2
Any tangent to the given circle will be ?? +2=?? (?? -1)+3v1+?? 2
??? =?? (?? -1)+
3v1+?? 2
-2
Q7. If the distances from the origin to the centres of three circles ?? ?? +?? ?? +?? ?? ?? ?? -?? ?? =?? (?? =
?? ,?? ,?? ) are in G.P. then the lengths of the tangents drawn to them from any point on the circle
?? ?? +?? ?? =?? ?? are in
(a) A.P.
(b) G.P .
(c) H.P.
(d) None of these
Ans : (b) The centres of the given circles are (-?? ?? ,0)(?? =1,2,3)
The distances from the origin to the centres are ?? ?? (?? =1,2,3). It is given that ?? 2
2
=?? 1
?? 3
.
Let ?? (h,?? ) be any point on the circle ?? 2
+?? 2
=?? 2
, then,
h
2
+?? 2
=?? 2
Now, ?? ?? = length of the tangent from (h,?? ) to ?? 2
+?? 2
+2?? ?? ?? -?? 2
=0
=vh
2
+?? 2
+2?? ?? h-?? 2
=v?? 2
+2?? ?? h-?? 2
=v2?? ?? h
[?h
2
+?? 2
=?? 2
and ?? =1,2,3]
Therefore, ?? 2
2
=2?? 2
h=2h(v?? 1
?? 3
)
[??? 2
2
=?? 1
?? 3
]
=v2h?? 1
v2h?? 3
=?? 1
?? 3
. Hence, ?? 1
,?? 2
,?? 3
are in G.P .
Q8. The angle between a pair of tangents drawn from a point ?? to the circle ?? ?? +?? ?? +?? ?? -?? ?? +
?? ?????? ?? ?? +???? ?????? ?? ?? =?? is
(a) ?? ?? +?? ?? +?? ?? -?? ?? +?? =??
(b) ?? ?? +?? ?? +?? ?? -?? ?? -?? =??
(c) ?? ?? +?? ?? +?? ?? -?? ?? -?? =??
(d) ?? ?? +?? ?? +?? ?? -?? ?? +?? =??
Ans : (d) The centre of the circle ?? 2
+?? 2
+4?? -6?? +9sin
2
?? +13cos
2
?? =0 is ?? (-2,3) and its
radius is
v2
2
+(-3)
2
-9sin
2
?? -13cos
2
?? =v4+9-9sin
2
?? -13cos
2
?? =2sin ??
Let ?? (h,?? ) be any point on the locus. The ??????? =?? . Also ??????? =?? /2 i.e. triangle ?????? is a right
angle triangle.
Thus sin ?? =
????
????
=
2sin ?? v(h+2)
2
+(?? -3)
2
?v(h+2)
2
+(?? -3)
2
=2?(h+2)
2
+(?? -3)
2
=4or h
2
+?? 2
+4h-6?? +9=0
Thus the required equation of the locus is ?? 2
+?? 2
+4?? -6?? +9=0.
Q9. Tangents are drawn from any point on the circle ?? ?? +?? ?? =?? ?? to the circle ?? ?? +?? ?? =?? ?? . If
the chord of contact touches the circle ?? ?? +?? ?? =?? ?? ,?? >?? , then
(a) a, b, c are in A.P .
(b) a, b, c are in G.P .
(c) a, b, c are in H.P . (d) a, c, b are in G.P .
Ans : (b) Chord of contact of any point (?? cos ?? ,?? sin ?? ) on 1
st
circle with respect to 2
nd
circle is
???? cos ?? +???? sin ?? =?? 2
This chord touches the circle ?? 2
+?? 2
=?? 2
,
Hence, Radius = Perpendicular distance of chord from centre.
?? =
?? 2
?? vcos
2
?? +sin
2
?? ??? 2
=???? .Hence ?? ,?? ,?? are in G.P .
Q10. A foot of the normal from the point (?? ,?? ) to a circle is (?? ,?? ) and a diameter of the circle has
the equation ?? ?? -?? =?? . Then the equation of the circle is
(a) ?? ?? +?? ?? +?? ?? -?? =??
(b) ?? ?? +?? ?? -?? ?? -?? =??
(c) ?? ?? +?? ?? -?? ?? -?? =??
(d) None of these.
Ans : (b) The line joining (4,3) and (2,1) is also along a diameter. So, the centre is the intersection of
the diameters 2?? -?? =2 and ?? -3=(?? -4). Solving these, the centre =(1,0)
? Radius = Distance between (1,0) and (2,1)=v2.
? Equation of circle (?? -1)
2
+?? 2
=(v2)
2
??? 2
+?? 2
-2?? -1=0
Q11. The polar of the point (?? ,-
?? ?? ) with respect to circle (?? -?? )
?? +?? ?? =?? is
(a) ?? ?? -???? ?? +?? =??
(b) ?? ?? -?? -???? =??
(c) ???? ?? -?? -???? =??
(d) ?? -???? ?? -?? =??
Ans : (b)
The polar of the point (5,-
1
2
) is ?? ?? 1
+?? ?? 1
+?? (?? +?? 1
)+?? (?? +?? 1
)+?? =0
?5?? -
1
2
?? -2(?? +5)+0+0=0?3?? -
?? 2
-10=0?6?? -?? -20=0.
Q12. The equation of the circle which touches the circle ?? ?? +?? ?? -?? ?? +?? ?? +???? =?? externally
and to which the lines ?? ?? -?? ???? -?? ?? +?? ?? =?? are normals, is
(a) ?? ?? +?? ?? -?? ?? -?? ?? -?? =??
(b) ?? ?? +?? ?? +?? ?? +?? ?? +?? =??
(c) ?? ?? +?? ?? -?? ?? -?? ?? +?? =??
(d) ?? ?? +?? ?? -?? ?? -?? ?? +?? =??
Ans : (d) Joint equations of normals are ?? 2
-3???? -3?? +9?? =0??? (?? -3?? )-3(?? -3?? )=0?
(?? -3)(?? -3?? )=0
? Given normals are ?? -3=0 and ?? -3?? =0, which intersect at centre of circle whose
coordinates are (3,1).
The given circle is ?? 1
=(3,-3),?? 1
=1;?? 2
=(3,1),?? 2
= ?
If the two circles touch externally, then ?? 1
?? 2
=?? 1
+?? 2
?4=1+?? 2
??? 2
=3
? Equation of required circle is (?? -3)
2
+(?? -1)
2
=(3)
2
??? 2
+?? 2
-6?? -2?? +1=0
Q13. There are two circles whose equations are ?? ?? +?? ?? =?? and ?? ?? +?? ?? -?? ?? -?? ?? +?? ?? =
?? ,?? ??? . If the two circles have exactly two common tangents, then the number of possible values
of ?? is
(a) 2
(b) 8
(c) 9
(d) None of these
Ans : (c) For ?? 2
+?? 2
=9, the centre =(0,0) and the radius =3
For ?? 2
+?? 2
-8?? -6?? +?? 2
=0. The centre =(4,3) and the radius =v(4)
2
+(3)
2
-?? 2
?4
2
+3
2
-?? 2
>0 or ?? 2
<5
2
or -5<?? <5.
Circles should cut to have exactly two common tangents.
So, ?? 1
+?? 2
>?? 1
?? 2
,?3+v25-?? 2
>v(4)
2
+(3)
2
or v25-?? 2
>2 or 25-?? 2
>4
??? 2
<21 or -v21<?? <v21
Therefore, common values of ?? should satisfy -v21<?? <v21.
But ?? ??? , So, ?? =-4,-3,……..3,4. ? Number of possible values of ?? =9.
Q14. If the circle ?? ?? +?? ?? =?? bisects the circumference of the circle ?? ?? +?? ?? -?? ?? +?? ?? +?? =?? ,
then ?? equals
(a) 4
(b) -4
(c) 16
(d) -16
Ans : (c) The common chord of given circles is ?? 1
-?? 2
=0?2?? -6?? -4-?? =0
Since, ?? 2
+?? 2
=4 bisects the circumferences of the circle ?? 2
+?? 2
-2?? +6?? +?? =0, therefore (i)
passes through the centre of second circle i.e. (1,-3). ? 2+18-4-?? =0??? =16.
Q15. A circle passes through the origin and has its centre on ?? =?? . If it cuts ?? ?? +?? ?? -?? ?? -?? ?? +
???? =?? orthogonally, then the equation of the circle is
(a) ?? ?? +?? ?? -?? -?? =??
(b) ?? ?? +?? ?? -?? ?? -?? ?? =??
(c) ?? ?? +?? ?? -?? ?? -?? ?? =??
(d) ?? ?? +?? ?? +?? ?? +?? ?? =??
Ans : (c) Let the required circle be ?? 2
+?? 2
+2???? +2???? +?? =0
This passes through (0,0), therefore ?? =0
The centre (-?? ,-?? ) of (i) lies on ?? =?? , hence ?? =?? .
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209 videos|443 docs|143 tests
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FAQs on Circle Solved Examples - Mathematics (Maths) for JEE Main & Advanced
| 1. How do you find the circumference of a circle? |  |
Ans. The circumference of a circle can be found using the formula: Circumference = 2 * π * radius.
| 2. What is the relationship between the circumference and diameter of a circle? | |
Ans. The circumference of a circle is directly proportional to its diameter, with the constant of proportionality being π (pi).
| 3. How is the area of a circle calculated? | |
Ans. The area of a circle can be calculated using the formula: Area = π * radius^2.
| 4. Can you find the radius of a circle if you know its circumference? | |
Ans. Yes, the radius of a circle can be found using the formula: radius = circumference / (2 * π).
| 5. How are the concepts of circles used in real-life applications? | |
Ans. Circles are used in various real-life applications such as designing wheels, gears, pulleys, and roundabouts.
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