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JEE • MATHEMATICS • SYSTEM OF CIRCLES

System of circles practice · Mixed difficulty

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Practice format and chapter quiz

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Showing 1-30 of 98 questions
1 Easy
1 of 98
Application +4 | -1
x=1x = 1 is the radical axis of the two orthogonally intersecting circles. If x2+y2=4{x^2} + {y^2} = 4 is one of the circles, then the other circle is
View answer and explanation
Correct option: B
x2+y2−8x+4=0{x^2} + {y^2} - 8x + 4 = 0
Option B (x2+y2−8x+4=0{x^2} + {y^2} - 8x + 4 = 0) is correct because it is orthogonal to the given circle and has x = 1 as the radical axis.
2 Easy
2 of 98
Application +4 | -1
The circle x2+y2+8y−4=0{x^2} + {y^2} + 8y - 4 = 0, cuts the real circle x2+y2+gx+4=0{x^2} + {y^2} + gx + 4 = 0 orthogonally, if
View answer and explanation
Correct option: A
For any real value of gg
Option A (For any real value of gg) is correct because the orthogonality condition is satisfied regardless of the value of g.
3 Easy
3 of 98
Application +4 | -1
The equation of the circle whose diameter is the common chord of the circles x2+y2+3x+2y+1=0x^2+y^2+3 x+2 y+1=0 and x2+y2+3x+4y+2=0x^2+y^2+3 x+4 y+2=0, is
View answer and explanation
Correct option: C
2x2+2y2+6x+2y+1=02 x^2+2 y^2+6 x+2 y+1=0
Option C (2x2+2y2+6x+2y+1=02 x^2+2 y^2+6 x+2 y+1=0) is correct because it represents the equation of the circle whose diameter is the common chord of the given circles.
4 Easy
4 of 98
Application +4 | -1
If the circles x2+y2+2ky+2x+6=0{x^2} + {y^2} + 2ky + 2x + 6 = 0 and x2+y2+2ky+k=0{x^2} + {y^2} + 2ky + k = 0 intersects orthogonally. Then, the value of kk is
View answer and explanation
Correct option: C
−32- \frac{3}{2}
Option C (−32- \frac{3}{2}) is correct because it is one of the values of k obtained by solving the quadratic equation derived from the orthogonality condition of the circles.
5 Easy
5 of 98
Analyzing +4 | -1
If the circles x2+y2+2ax+cy+a=0x^2+y^2+2 a x+c y+a=0 and x2+y2−3ax+dy−1=0x^2+y^2-3 a x+d y-1=0 intersect in two distinct points PP and QQ, then the line 5x+by−a=05 x+b y-a=0 passes through PP and QQ for
View answer and explanation
Correct option: B
no value of aa
Option B (no value of aa) is correct because the condition for the line 5x+by−a=05x + by - a = 0 to pass through the intersection points P and Q leads to a quadratic equation with no real solutions for aa.
6 Easy
6 of 98
Analyzing +4 | -1
The equation of the circle having its centre on the line x+2y−3=0x + 2y - 3 = 0 and passing through the point of intersection of the circles x2+y2−4y+1=0{x^2} + {y^2} - 4y + 1 = 0 and x2+y2−4x−2y+4=0{x^2} + {y^2} - 4x - 2y + 4 = 0 is
View answer and explanation
Correct option: A
x2+y2−6x+7=0{x^2} + {y^2} - 6x + 7 = 0
The correct answer is x2+y2−6x+7=0{x^2} + {y^2} - 6x + 7 = 0 (Option A) because it satisfies the condition of passing through the intersection of the given circles and having its center on the line x+2y−3=0x + 2y - 3 = 0.
7 Medium
7 of 98
Application +4 | -1
If a circle passes through the point (1, 2) and cuts the circle x2+y2=4{x^2} + {y^2} = 4 orthogonally, then the equation of the locus of its centre is
View answer and explanation
Correct option: C
2x+4y−9=02x + 4y - 9 = 0
Option C (2x+4y−9=02x + 4y - 9 = 0) is correct because it represents the locus of the center of a circle passing through (1, 2) and intersecting the given circle orthogonally.
8 Medium
8 of 98
Application +4 | -1
The equation of the circle which passes through the origin and cuts orthogonally each of the circles x2+y2−6x+8=0{x^2} + {y^2} - 6x + 8 = 0 and x2+y2−2x−2y−7=0{x^2} + {y^2} - 2x - 2y - 7 = 0 is
View answer and explanation
Correct option: B
3x2+3y2−8x+29y=03{x^2} + 3{y^2} - 8x + 29y = 0
Option C ((7,−5)(7, - 5)) is correct because it represents the other end of the diameter, calculated using the midpoint formula with the given point and the center of the circle.
9 Easy
9 of 98
Application +4 | -1
The length of the common chord of the circles x2+y2+4x+1=0{x^2} + {y^2} + 4x + 1 = 0 and x2+y2+4y−1=0{x^2} + {y^2} + 4y - 1 = 0
View answer and explanation
Correct option: A
152\sqrt {\frac{{15}}{2}}
Option A (152\sqrt {\frac{{15}}{2}}) is correct because it represents the length of the common chord of the two circles.
10 Easy
10 of 98
Application +4 | -1
The equation of circle concentric with circle x2+y2−6x−4y+1=0x^2+y^2-6 x-4 y+1=0 and passing through origin is
View answer and explanation
Correct option: A
x2+y2−6x−4y=0x^2+y^2-6 x-4 y=0
Option A (x2+y2−6x−4y=0x^2+y^2-6 x-4 y=0) is correct because it has the same center as the given circle and passes through the origin.
11 Medium
11 of 98
Application +4 | -1
If a chord of a circle x2+y2=25x^2+y^2=25 with one extremity at (4,3)(4,3) subtends a right angle at the centre of this circle, then the coordinates of the other extremity of this chord can be.
View answer and explanation
Correct option: D
(−3,4)(-3,4)
Option D ((-3,4)) is correct because the dot product of the position vectors of the two extremities of the chord is zero, indicating a right angle at the center.
12 Easy
12 of 98
Analyzing +4 | -1
The radical axis of the circles x2+y2+2gx+2fy+c=0x^2+y^2+2 g x+2 f y+c=0 and 2(x2+y2)+3x+2y+2c=02\left(x^2+y^2\right)+3 x+2 y+2 c=0 touches the circle x2+y2+2x+2y+1=0x^2+y^2+2 x+2 y+1=0, if
View answer and explanation
Correct option: B
g=43g=\frac{4}{3} and f=12f=\frac{1}{2}
Option B (g=43g=\frac{4}{3} and f=12f=\frac{1}{2}) is correct because these values satisfy the condition for the radical axis to touch the third circle.
13 Easy
13 of 98
Analyzing +4 | -1
The circles x2+y2+x+y=0{x^2} + {y^2} + x + y = 0 and x2+y2+x−y{x^2} + {y^2} + x - y intersect at an angle of
View answer and explanation
Correct option: D
π2\frac{\pi }{2}
Option D π2\frac{\pi }{2} is correct because the angle of intersection between the two circles is 90 degrees, indicating they are orthogonal.
14 Hard
14 of 98
Analyzing +4 | -1
If the circle C1  :x2+y2=16{C_1}\;:{x^2} + {y^2} = 16 intersects another circle C2{C_2} of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/43/4, the coordinates of the centre of C2{C_2} are
View answer and explanation
Correct option: A
(−9/5,12/5),(9/5,−12/5)\left( { - 9/5,12/5} \right),\left( {9/5, - 12/5} \right)
Option D is correct because the condition for no solution is that the constant 'c' must be greater than a certain value derived from tangency.
15 Easy
15 of 98
Application +4 | -1
If a circle passes through the point (a,b)\left( {a,b} \right) and cuts the circle x2+y2=p2{x^2} + {y^2} = {p^2} orthogonally, then the equation of the locus of its centre is
View answer and explanation
Correct option: A
2ax+2by−(a2+b2+p2)=02ax + 2by - \left( {{a^2} + {b^2} + {p^2}} \right) = 0
The correct answer is A (2ax+2by−(a2+b2+p2)=02ax + 2by - \left( {{a^2} + {b^2} + {p^2}} \right) = 0) because it represents the locus of the center of the circle passing through (a, b) and cutting the given circle orthogonally.

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16 Easy
16 of 98
Application +4 | -1
If a circle passes through the point (a,b)(a, b) and cuts the circle x2+y2=p2x^2+y^2=p^2 orthogonally, then the equation of the locus of its centre is
View answer and explanation
Correct option: D
2ax+2by−(a2+b2+p2)=02 a x+2 b y-\left(a^2+b^2+p^2\right)=0
Option D (2ax+2by−(a2+b2+p2)=02 a x+2 b y-\left(a^2+b^2+p^2\right)=0) is correct because it represents the locus of the center of the circle that passes through (a, b) and cuts the circle x2+y2=p2x^2+y^2=p^2 orthogonally.
17 Easy
17 of 98
Analyzing +4 | -1
The angle of intersection of the circles x2+y2=4{x^2} + {y^2} = 4 and x2+y2=2x+2y,{x^2} + {y^2} = 2x + 2y, is
View answer and explanation
Correct option: D
π4\frac{\pi }{4}
Option D (π4\frac{\pi }{4}) is correct because it is the angle between the circles, calculated using the radii and the distance between their centers.
18 Medium
18 of 98
Analyzing +4 | -1
The circle x2+y2=4{x^2} + {y^2} = 4 cuts the circle x2+y2−2x−4=0{x^2} + {y^2} - 2x - 4 = 0 at the points AA and BB. If the circle x2+y2−4x−k=0{x^2} + {y^2} - 4x - k = 0 passes through AA and BB then the value of kk is
View answer and explanation
Correct option: D
4
Option D (4) is correct because by substituting the intersection points of the first two circles into the third circle's equation, we find that k equals 4.
19 Easy
19 of 98
Application +4 | -1
The equation of the circle whose diameter is the common chord of the circles x2+y2+2x+3y+2=0{x^2} + {y^2} + 2x + 3y + 2 = 0 and x2+y2+2x−3y−4=0{x^2} + {y^2} + 2x - 3y - 4 = 0 is
View answer and explanation
Correct option: C
x2+y2+2x+2y+1=0{x^2} + {y^2} + 2x + 2y + 1 = 0
Option C (x2+y2+2x+2y+1=0{x^2} + {y^2} + 2x + 2y + 1 = 0) is correct because it represents the circle whose diameter is the common chord of the given circles.
20 Easy
20 of 98
Analyzing +4 | -1
x2+y2−6x−6y+4=0,x2+y2−2x−4y+3=0,{x^2} + {y^2} - 6x - 6y + 4 = 0,{x^2} + {y^2} - 2x - 4y + 3 = 0, x2+y2+2kx+2y+1=0{x^2} + {y^2} + 2kx + 2y + 1 = 0. If the radical centre of the above three circles exists, then which of the following cannot be the value of kk
View answer and explanation
Correct option: C
5
Option C (5) is correct because the radical center exists only when the determinant of the coefficients of the radical axes is non-zero, leading to k≠5k \ne 5.
21 Easy
21 of 98
Application +4 | -1
If the circle x2+y2+2gx+2fy+c=0x^2+y^2+2 g x+2 f y+c=0 bisects the circumference of the circle x2+y2+2g′x+2f′y+c′=0x^2+y^2+2 g^{\prime} x+2 f^{\prime} y+c^{\prime}=0, then the length of the common chord of these two circles, is
View answer and explanation
Correct option: B
2g′2+f′2−c′2 \sqrt{g^{\prime 2}+f^{\prime 2}-c^{\prime}}
Option B (2g′2+f′2−c′2 \sqrt{g^{\prime 2}+f^{\prime 2}-c^{\prime}}) is correct because when one circle bisects the circumference of another, the common chord is the diameter of the bisected circle.
22 Easy
22 of 98
Application +4 | -1
The centre of the circle, which cuts orthogonally each of the three circles x2+y2+2x+17y+4=0{x^2} + {y^2} + 2x + 17y + 4 = 0 and x2+y2+7x+6y+11=0,x2+y2−x+22y+3=0{x^2} + {y^2} + 7x + 6y + 11 = 0,{x^2} + {y^2} - x + 22y + 3 = 0, is
View answer and explanation
Correct option: A
(3,2)\left( {3,2} \right)
Option A (3,2) is correct because it represents the center of the circle that intersects the three given circles orthogonally, found by solving the equations derived from the orthogonality condition.
23 Easy
23 of 98
Analyzing +4 | -1
A circle CC touches the line x=2yx = 2y at the point (2, 1) and intersects the circle C1:x2+y2+2y−5=0{C_1}:{x^2} + {y^2} + 2y - 5 = 0 at two points PP and QQ such that PQPQ is a diameter of C1{C_1}. Then the diameter of CC is
View answer and explanation
Correct option: A
757\sqrt 5
Option A (757\sqrt 5) is correct because it is the diameter of the circle C, which is found by using the condition that the common chord PQ is a diameter of circle C1.
24 Medium
24 of 98
Analyzing +4 | -1
If the circles x2+y2+5Kx+2y+K=0{x^2} + {y^2} + 5Kx + 2y + K = 0 and 2(x2+y2)+2Kx+3y−1=0,(K∈R),2\left( {{x^2} + {y^2}} \right) + 2Kx + 3y - 1 = 0,\left( {K \in R} \right), intersect at the points PP and Q,Q, then the line 4x+5y−K=04x + 5y - K = 0 passes through PP and Q,Q, for
View answer and explanation
Correct option: A
No value of K
Option A (No value of K) is correct because the condition for the line to pass through the intersection points of the circles leads to conflicting values of K.
25 Easy
25 of 98
Analyzing +4 | -1
A circle SS passes through the point (0, 1) and is orthogonal to the circles (x−1)2+y2=16(x-1)^2+y^2=16 and x2+y2=1x^2+y^2=1. Then,
View answer and explanation
Correct option: B
Radius of SS is 7
The correct answer is B (Radius of SS is 7) because after finding the center and using the formula for radius, the radius is calculated to be 7.
26 Hard
26 of 98
Application +4 | -1
One of the limit point of the coaxial system of circles containing x2+y2−6  x−6  y+4=0,    x2+y2−2  x−4  y+3=0,{x^2} + {y^2} - 6\;x - 6\;y + 4 = 0,\;\;{x^2} + {y^2} - 2\;x - 4\;y + 3 = 0, is
View answer and explanation
Correct option: A
(−1,1)\left( { - 1,1} \right)
Option A ((−1,1)\left( { - 1,1} \right)) is correct because it represents one of the limit points of the coaxial system of circles, found by setting the radius to zero.
27 Easy
27 of 98
Application +4 | -1
The radical axis of the coaxial system of circles with limiting points (1,2)\left( {1,2} \right) and (−2,1)\left( { - 2,1} \right) is
View answer and explanation
Correct option: B
3x+y=03x + y = 0
Option B (3x+y=03x + y = 0) is correct because it represents the equation of the radical axis derived from the limiting points of the coaxial system.
28 Medium
28 of 98
Application +4 | -1
If the circle x2+y2+4x+22y+c=0    {x^2} + {y^2} + 4x + 22y + c = 0\;\; bisects the circumference of the circle x2+y2−2x+8y−d=0{x^2} + {y^2} - 2x + 8y - d = 0, then c+dc + d is equal to
View answer and explanation
Correct option: B
50
Option B (50) is correct because the sum c+d is found by using the fact that the common chord passes through the center of the second circle.
29 Easy
29 of 98
Analyzing +4 | -1
If PP and QQ are the points of intersection of the circles x2+y2+3x+7y+2p−5=0x^2+y^2+3 x+7 y+2 p-5=0 and x2+y2+2x+2y−p2=0x^2+y^2+2 x+2 y-p^2=0 then there is a circle passing through P,QP, Q and (1,1)(1,1) for :
View answer and explanation
Correct option: A
all except one value of pp
Option A (all except one value of pp) is correct because the equation for λ\lambda is undefined when p=−1p = -1, indicating that there is no circle passing through P, Q, and (1,1) for that specific value of pp.
30 Hard
30 of 98
Application +4 | -1
Origin is a limiting point of a coaxial system of which x2+y2−6x−8y+1=0{x^2} + {y^2} - 6x - 8y + 1 = 0 is a member. The other limiting point is
View answer and explanation
Correct option: B
(3/25,4/25)\left( {3/25,4/25} \right)
Option B ((3/25,4/25)\left( {3/25,4/25} \right)) is correct because it is calculated using the formula for limiting points of a coaxial system of circles.
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