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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 availableRead More
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