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Circles: JEE Mains Previous Year Questions (2021-2024) | Mathematics (Maths) for JEE Main & Advanced PDF Download

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JEE Mains Previous Year Questions 
(2021-2024): Circle 
2024 
Q1: Let the locus of the midpoints of the chords of the circle ?? ?? +(?? -?? )
?? =?? drawn from the origin 
intersect the line ?? +?? =?? at ?? and ?? . Then, the length of ???? is : 
A. 
?? ?? 
B. 1 
C. 
?? v?? 
D. v??    [JEE Main 2024 (Online) 1st February Evening Shift] 
Ans: (c) 
Let mid-point is (?? ,?? ) 
?? 2
+?? 2
-2?? =0
?? ?? 1
+?? ?? 1
-(?? +?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 
It is passing through origin 
 So, 0+0-(0+?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 ?-?? 1
=?? 1
2
+?? 1
2
-2?? 1
 ??? 1
2
+?? 1
2
-?? 1
=0
?? 2
+?? 2
-?? =0
 
? It intersects the line ?? +?? =1 
So put ?? =1-?? in equation (1) 
 (1-?? )
2
+?? 2
-?? =0
2?? 2
-3?? +1=0
 ?(?? -1)(2?? -1)=0
 ??? =1,
1
2
 
??? (0,1) and ?? (
1
2
,
1
2
) 
So, ???? =
v
(
1
2
-0)
2
+(
1
2
-1)
2
=
1
v2
 
Q2: Let ?? :?? ?? +?? ?? =?? and ?? '
:?? ?? +?? ?? -?? ???? +?? =?? be two circles. If the set of all values of ?? so 
that the circles ?? and ?? intersect at two distinct points, is ?? -[?? ,?? ], then the point (???? +???? ,???? ?? -
???? ) lies on the curve : 
A. ?? ?? +?? ?? ?? -?? ?? +?? ?? =?? 
B. ?? ?? ?? -?? =-???? 
Page 2


JEE Mains Previous Year Questions 
(2021-2024): Circle 
2024 
Q1: Let the locus of the midpoints of the chords of the circle ?? ?? +(?? -?? )
?? =?? drawn from the origin 
intersect the line ?? +?? =?? at ?? and ?? . Then, the length of ???? is : 
A. 
?? ?? 
B. 1 
C. 
?? v?? 
D. v??    [JEE Main 2024 (Online) 1st February Evening Shift] 
Ans: (c) 
Let mid-point is (?? ,?? ) 
?? 2
+?? 2
-2?? =0
?? ?? 1
+?? ?? 1
-(?? +?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 
It is passing through origin 
 So, 0+0-(0+?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 ?-?? 1
=?? 1
2
+?? 1
2
-2?? 1
 ??? 1
2
+?? 1
2
-?? 1
=0
?? 2
+?? 2
-?? =0
 
? It intersects the line ?? +?? =1 
So put ?? =1-?? in equation (1) 
 (1-?? )
2
+?? 2
-?? =0
2?? 2
-3?? +1=0
 ?(?? -1)(2?? -1)=0
 ??? =1,
1
2
 
??? (0,1) and ?? (
1
2
,
1
2
) 
So, ???? =
v
(
1
2
-0)
2
+(
1
2
-1)
2
=
1
v2
 
Q2: Let ?? :?? ?? +?? ?? =?? and ?? '
:?? ?? +?? ?? -?? ???? +?? =?? be two circles. If the set of all values of ?? so 
that the circles ?? and ?? intersect at two distinct points, is ?? -[?? ,?? ], then the point (???? +???? ,???? ?? -
???? ) lies on the curve : 
A. ?? ?? +?? ?? ?? -?? ?? +?? ?? =?? 
B. ?? ?? ?? -?? =-???? 
C. ?? ?? -?? ?? ?? =?? 
D. ?? ?? ?? +?? ?? =????   [JEE Main 2024 (Online) 1st February Morning Shift] 
Ans: (d) 
Explanation: Currently no explanation available 
Q3: Let a variable line passing through the centre of the circle ?? ?? +?? ?? -???? ?? -?? ?? =?? , meet the 
positive co-ordinate axes at the points ?? and ?? . Then the minimum value of ???? +???? , where ?? is the 
origin, is equal to 
A. 12 
B. 20 
C. 24 
D. 18   [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: (d) 
 (?? -2)=?? (?? -8)
 ??? -intercept 
 ?(
-2
?? +8)
 ??? -intercept 
 ?(-8 m+2)
 ?OA+OB=
-2
 m
+8-8 m+2
f
'
(m)=
2
 m
2
-8=0
 ?m
2
=
1
4
 ?m=
-1
2
 ?f(
-1
2
)=18
 ?Minimum=18
 
Q4: If one of the diameters of the circle ?? ?? +?? ?? -???? ?? +?? ?? +???? =?? is a chord of another circle ?? , 
whose center is the point of intersection of the lines ?? ?? +?? ?? =???? and ?? ?? -?? ?? =?? , then the radius 
of the circle ?? is : 
A. 4 
B. ?? v?? 
C. 6 
D. v????    [JEE Main 2024 (Online) 31st January Morning Shift] 
Ans: (c) 
Page 3


JEE Mains Previous Year Questions 
(2021-2024): Circle 
2024 
Q1: Let the locus of the midpoints of the chords of the circle ?? ?? +(?? -?? )
?? =?? drawn from the origin 
intersect the line ?? +?? =?? at ?? and ?? . Then, the length of ???? is : 
A. 
?? ?? 
B. 1 
C. 
?? v?? 
D. v??    [JEE Main 2024 (Online) 1st February Evening Shift] 
Ans: (c) 
Let mid-point is (?? ,?? ) 
?? 2
+?? 2
-2?? =0
?? ?? 1
+?? ?? 1
-(?? +?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 
It is passing through origin 
 So, 0+0-(0+?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 ?-?? 1
=?? 1
2
+?? 1
2
-2?? 1
 ??? 1
2
+?? 1
2
-?? 1
=0
?? 2
+?? 2
-?? =0
 
? It intersects the line ?? +?? =1 
So put ?? =1-?? in equation (1) 
 (1-?? )
2
+?? 2
-?? =0
2?? 2
-3?? +1=0
 ?(?? -1)(2?? -1)=0
 ??? =1,
1
2
 
??? (0,1) and ?? (
1
2
,
1
2
) 
So, ???? =
v
(
1
2
-0)
2
+(
1
2
-1)
2
=
1
v2
 
Q2: Let ?? :?? ?? +?? ?? =?? and ?? '
:?? ?? +?? ?? -?? ???? +?? =?? be two circles. If the set of all values of ?? so 
that the circles ?? and ?? intersect at two distinct points, is ?? -[?? ,?? ], then the point (???? +???? ,???? ?? -
???? ) lies on the curve : 
A. ?? ?? +?? ?? ?? -?? ?? +?? ?? =?? 
B. ?? ?? ?? -?? =-???? 
C. ?? ?? -?? ?? ?? =?? 
D. ?? ?? ?? +?? ?? =????   [JEE Main 2024 (Online) 1st February Morning Shift] 
Ans: (d) 
Explanation: Currently no explanation available 
Q3: Let a variable line passing through the centre of the circle ?? ?? +?? ?? -???? ?? -?? ?? =?? , meet the 
positive co-ordinate axes at the points ?? and ?? . Then the minimum value of ???? +???? , where ?? is the 
origin, is equal to 
A. 12 
B. 20 
C. 24 
D. 18   [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: (d) 
 (?? -2)=?? (?? -8)
 ??? -intercept 
 ?(
-2
?? +8)
 ??? -intercept 
 ?(-8 m+2)
 ?OA+OB=
-2
 m
+8-8 m+2
f
'
(m)=
2
 m
2
-8=0
 ?m
2
=
1
4
 ?m=
-1
2
 ?f(
-1
2
)=18
 ?Minimum=18
 
Q4: If one of the diameters of the circle ?? ?? +?? ?? -???? ?? +?? ?? +???? =?? is a chord of another circle ?? , 
whose center is the point of intersection of the lines ?? ?? +?? ?? =???? and ?? ?? -?? ?? =?? , then the radius 
of the circle ?? is : 
A. 4 
B. ?? v?? 
C. 6 
D. v????    [JEE Main 2024 (Online) 31st January Morning Shift] 
Ans: (c) 
 
 
2?? +3?? =12
3?? -2?? =5
13?? =39
?? =3,?? =2
 
Center of given circle is (5,-2) 
Radius v25+4-13=4 
 ?CM=v4+16=5v2
 ?CP=v16+20=6
 
Q5: If the circles (?? +?? )
?? +(?? +?? )
?? =?? ?? and ?? ?? +?? ?? -?? ?? -?? ?? +?? =?? intersect at exactly two 
distinct points, then 
A. 
?? ?? <?? <?? 
B. ?? <?? <?? 
C. ?? <?? <?? 
D. ?? < r <??    [JEE Main 2024 (Online) 30th January Morning Shift] 
Ans: (b) 
If two circles intersect at two distinct points 
 ?|r
1
-r
2
|<C
1
C
2
<r
1
+r
2
 |r-2|<v9+16<r+2
 |r-2|<5 and r+2>5
 -5<r-2<5 r>3……..2
 -3<r<7
 
From (1) and (2) 
3<r<7 
 
Page 4


JEE Mains Previous Year Questions 
(2021-2024): Circle 
2024 
Q1: Let the locus of the midpoints of the chords of the circle ?? ?? +(?? -?? )
?? =?? drawn from the origin 
intersect the line ?? +?? =?? at ?? and ?? . Then, the length of ???? is : 
A. 
?? ?? 
B. 1 
C. 
?? v?? 
D. v??    [JEE Main 2024 (Online) 1st February Evening Shift] 
Ans: (c) 
Let mid-point is (?? ,?? ) 
?? 2
+?? 2
-2?? =0
?? ?? 1
+?? ?? 1
-(?? +?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 
It is passing through origin 
 So, 0+0-(0+?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 ?-?? 1
=?? 1
2
+?? 1
2
-2?? 1
 ??? 1
2
+?? 1
2
-?? 1
=0
?? 2
+?? 2
-?? =0
 
? It intersects the line ?? +?? =1 
So put ?? =1-?? in equation (1) 
 (1-?? )
2
+?? 2
-?? =0
2?? 2
-3?? +1=0
 ?(?? -1)(2?? -1)=0
 ??? =1,
1
2
 
??? (0,1) and ?? (
1
2
,
1
2
) 
So, ???? =
v
(
1
2
-0)
2
+(
1
2
-1)
2
=
1
v2
 
Q2: Let ?? :?? ?? +?? ?? =?? and ?? '
:?? ?? +?? ?? -?? ???? +?? =?? be two circles. If the set of all values of ?? so 
that the circles ?? and ?? intersect at two distinct points, is ?? -[?? ,?? ], then the point (???? +???? ,???? ?? -
???? ) lies on the curve : 
A. ?? ?? +?? ?? ?? -?? ?? +?? ?? =?? 
B. ?? ?? ?? -?? =-???? 
C. ?? ?? -?? ?? ?? =?? 
D. ?? ?? ?? +?? ?? =????   [JEE Main 2024 (Online) 1st February Morning Shift] 
Ans: (d) 
Explanation: Currently no explanation available 
Q3: Let a variable line passing through the centre of the circle ?? ?? +?? ?? -???? ?? -?? ?? =?? , meet the 
positive co-ordinate axes at the points ?? and ?? . Then the minimum value of ???? +???? , where ?? is the 
origin, is equal to 
A. 12 
B. 20 
C. 24 
D. 18   [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: (d) 
 (?? -2)=?? (?? -8)
 ??? -intercept 
 ?(
-2
?? +8)
 ??? -intercept 
 ?(-8 m+2)
 ?OA+OB=
-2
 m
+8-8 m+2
f
'
(m)=
2
 m
2
-8=0
 ?m
2
=
1
4
 ?m=
-1
2
 ?f(
-1
2
)=18
 ?Minimum=18
 
Q4: If one of the diameters of the circle ?? ?? +?? ?? -???? ?? +?? ?? +???? =?? is a chord of another circle ?? , 
whose center is the point of intersection of the lines ?? ?? +?? ?? =???? and ?? ?? -?? ?? =?? , then the radius 
of the circle ?? is : 
A. 4 
B. ?? v?? 
C. 6 
D. v????    [JEE Main 2024 (Online) 31st January Morning Shift] 
Ans: (c) 
 
 
2?? +3?? =12
3?? -2?? =5
13?? =39
?? =3,?? =2
 
Center of given circle is (5,-2) 
Radius v25+4-13=4 
 ?CM=v4+16=5v2
 ?CP=v16+20=6
 
Q5: If the circles (?? +?? )
?? +(?? +?? )
?? =?? ?? and ?? ?? +?? ?? -?? ?? -?? ?? +?? =?? intersect at exactly two 
distinct points, then 
A. 
?? ?? <?? <?? 
B. ?? <?? <?? 
C. ?? <?? <?? 
D. ?? < r <??    [JEE Main 2024 (Online) 30th January Morning Shift] 
Ans: (b) 
If two circles intersect at two distinct points 
 ?|r
1
-r
2
|<C
1
C
2
<r
1
+r
2
 |r-2|<v9+16<r+2
 |r-2|<5 and r+2>5
 -5<r-2<5 r>3……..2
 -3<r<7
 
From (1) and (2) 
3<r<7 
 
Q6: Four distinct points (?? ?? ,?? ?? ),(?? ,?? ),(?? ,?? ) and (?? ,?? ) lie on a circle for ?? equal to: 
A. 
?? ????
 
B. 
?? ????
 
C. 
?? ????
 
D. 
?? ????
 
Ans: (c) 
Explanation: Currently no explanation available 
Q7: Consider two circles ?? ?? :?? ?? +?? ?? =???? and ?? ?? :(?? -?? )
?? +?? ?? =???? , where ?? ?(?? ,?? ) . Let the 
angle between the two radii (one to each circle) drawn from one of the intersection points of ?? ?? and 
?? ?? be ?????? -?? (
v????
?? ). If the length of common chord of ?? ?? and ?? ?? is ?? , then the value of (???? )
?? equals:      
[JEE Main 2024 (Online) 30th January Evening Shift] 
Ans: 1575 
?? 1
:?? 2
+?? 2
=25,?? 2
:(?? -?? )
2
+?? 2
=16
5<?? <9
 
 
?? =sin
-1
 (
v63
8
)
sin ?? =
v63
8
 
Area of ?OAP=
1
2
×?? (
?? 2
)=
1
2
×5×4sin ?? 
? ???? =40×
v63
8
???? =5×v63
 (???? )
2
=25×63=1575
 
Page 5


JEE Mains Previous Year Questions 
(2021-2024): Circle 
2024 
Q1: Let the locus of the midpoints of the chords of the circle ?? ?? +(?? -?? )
?? =?? drawn from the origin 
intersect the line ?? +?? =?? at ?? and ?? . Then, the length of ???? is : 
A. 
?? ?? 
B. 1 
C. 
?? v?? 
D. v??    [JEE Main 2024 (Online) 1st February Evening Shift] 
Ans: (c) 
Let mid-point is (?? ,?? ) 
?? 2
+?? 2
-2?? =0
?? ?? 1
+?? ?? 1
-(?? +?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 
It is passing through origin 
 So, 0+0-(0+?? 1
)=?? 1
2
+?? 1
2
-2?? 1
 ?-?? 1
=?? 1
2
+?? 1
2
-2?? 1
 ??? 1
2
+?? 1
2
-?? 1
=0
?? 2
+?? 2
-?? =0
 
? It intersects the line ?? +?? =1 
So put ?? =1-?? in equation (1) 
 (1-?? )
2
+?? 2
-?? =0
2?? 2
-3?? +1=0
 ?(?? -1)(2?? -1)=0
 ??? =1,
1
2
 
??? (0,1) and ?? (
1
2
,
1
2
) 
So, ???? =
v
(
1
2
-0)
2
+(
1
2
-1)
2
=
1
v2
 
Q2: Let ?? :?? ?? +?? ?? =?? and ?? '
:?? ?? +?? ?? -?? ???? +?? =?? be two circles. If the set of all values of ?? so 
that the circles ?? and ?? intersect at two distinct points, is ?? -[?? ,?? ], then the point (???? +???? ,???? ?? -
???? ) lies on the curve : 
A. ?? ?? +?? ?? ?? -?? ?? +?? ?? =?? 
B. ?? ?? ?? -?? =-???? 
C. ?? ?? -?? ?? ?? =?? 
D. ?? ?? ?? +?? ?? =????   [JEE Main 2024 (Online) 1st February Morning Shift] 
Ans: (d) 
Explanation: Currently no explanation available 
Q3: Let a variable line passing through the centre of the circle ?? ?? +?? ?? -???? ?? -?? ?? =?? , meet the 
positive co-ordinate axes at the points ?? and ?? . Then the minimum value of ???? +???? , where ?? is the 
origin, is equal to 
A. 12 
B. 20 
C. 24 
D. 18   [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: (d) 
 (?? -2)=?? (?? -8)
 ??? -intercept 
 ?(
-2
?? +8)
 ??? -intercept 
 ?(-8 m+2)
 ?OA+OB=
-2
 m
+8-8 m+2
f
'
(m)=
2
 m
2
-8=0
 ?m
2
=
1
4
 ?m=
-1
2
 ?f(
-1
2
)=18
 ?Minimum=18
 
Q4: If one of the diameters of the circle ?? ?? +?? ?? -???? ?? +?? ?? +???? =?? is a chord of another circle ?? , 
whose center is the point of intersection of the lines ?? ?? +?? ?? =???? and ?? ?? -?? ?? =?? , then the radius 
of the circle ?? is : 
A. 4 
B. ?? v?? 
C. 6 
D. v????    [JEE Main 2024 (Online) 31st January Morning Shift] 
Ans: (c) 
 
 
2?? +3?? =12
3?? -2?? =5
13?? =39
?? =3,?? =2
 
Center of given circle is (5,-2) 
Radius v25+4-13=4 
 ?CM=v4+16=5v2
 ?CP=v16+20=6
 
Q5: If the circles (?? +?? )
?? +(?? +?? )
?? =?? ?? and ?? ?? +?? ?? -?? ?? -?? ?? +?? =?? intersect at exactly two 
distinct points, then 
A. 
?? ?? <?? <?? 
B. ?? <?? <?? 
C. ?? <?? <?? 
D. ?? < r <??    [JEE Main 2024 (Online) 30th January Morning Shift] 
Ans: (b) 
If two circles intersect at two distinct points 
 ?|r
1
-r
2
|<C
1
C
2
<r
1
+r
2
 |r-2|<v9+16<r+2
 |r-2|<5 and r+2>5
 -5<r-2<5 r>3……..2
 -3<r<7
 
From (1) and (2) 
3<r<7 
 
Q6: Four distinct points (?? ?? ,?? ?? ),(?? ,?? ),(?? ,?? ) and (?? ,?? ) lie on a circle for ?? equal to: 
A. 
?? ????
 
B. 
?? ????
 
C. 
?? ????
 
D. 
?? ????
 
Ans: (c) 
Explanation: Currently no explanation available 
Q7: Consider two circles ?? ?? :?? ?? +?? ?? =???? and ?? ?? :(?? -?? )
?? +?? ?? =???? , where ?? ?(?? ,?? ) . Let the 
angle between the two radii (one to each circle) drawn from one of the intersection points of ?? ?? and 
?? ?? be ?????? -?? (
v????
?? ). If the length of common chord of ?? ?? and ?? ?? is ?? , then the value of (???? )
?? equals:      
[JEE Main 2024 (Online) 30th January Evening Shift] 
Ans: 1575 
?? 1
:?? 2
+?? 2
=25,?? 2
:(?? -?? )
2
+?? 2
=16
5<?? <9
 
 
?? =sin
-1
 (
v63
8
)
sin ?? =
v63
8
 
Area of ?OAP=
1
2
×?? (
?? 2
)=
1
2
×5×4sin ?? 
? ???? =40×
v63
8
???? =5×v63
 (???? )
2
=25×63=1575
 
Q8: Equations of two diameters of a circle are ?? ?? -?? ?? =?? and ?? ?? -?? ?? =?? . The line joining the 
points (-
????
?? ,-?? ) and (-
?? ?? ,?? ) intersects the circle at only one point ?? (?? ,?? ) . Then, ???? ?? -?? is equal 
to      [JEE Main 2024 (Online) 29th January Morning Shift] 
Ans: 2 
Explanation: Currently no explanation available 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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