Page 1
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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