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WorksheetsSystem of Circles Quiz
Total questions: 122
Worksheet time: 2hrs 1mins
A set of circles is said to be a system of circles if it contains at least two circles.
If d=r1+r2 circles touch each other externally.
If d>r1+r2 circles do not touch each other.
If d=|r1-r2| circles touch each other internally.
If d
If d=0 concentric circles.
If d<|r1-r2| one circle completely lies inside the other circles.
If two circles S=0 & S'=0 intersect at 'p', then the angle between the tangents of the two circles at p is called the angle between the circles at p.
If 'd' is the distance between the centres of two intersecting circles with radii r1,r2 and 'θ' is the angle between the circles then Cosθ = (d^2 - r1^2 - r2^2) / (2*r1*r2).
The condition that the two circles S=x^2+y^2+2gx+2fy+c=0 & S=x^2+y^2+2g'x+2f'y+c'=0 may cut each other orthogonally is 2gg'+2ff'=c+c'.
The point of intersection of direct common tangents of two circles is called as external centre of similitude.
If r1,r2 are radii of circles c1 and c2 then external centre of similitude divides the line joining c1,c2 in the ratio r1:r2 externally.
The point of intersection of transverse common tangents of two circles is called as internal centre of similitude.
If r1,r2 are radii of circles c1 and c2 then internal centre of similitude divides the line joining c1,c2 in the ratio r1:r2 internally.
If the radii of two circles are equal then the external centre of similitude does not exist.
If two circles are given then the locus of the points whose powers with respect to these two circles are equal is called as radical axis of these two circles.
If S=0, S'=0 are two circles in standard form then equation of radical axis is S-S'=0.
R.A is a straight line.
If two circles intersect in A and B, then their common chord is R.A i.e AB is R.A.
If two circles touch each other, then the common tangent at the point of contact is R.A.
The minimum number of R.A is zero when two circles are concentric.
If one circle lies in the other, then radical axis lies outside of both the circles.
If centres of three circles are non-collinear, then the point of concurrence of the three R.A is called a Radical centre(R.C).
If in a system of circles, each pair of circles have the same R.A then the system is called as a co-axial system of circles.
Simplest form of co-axial system of circles is x^2+y^2+2λx+c=0 (λ parameter, c is a constant).
The point circles of a Co-axial system of circles are called as limiting points of that co-axial system.
In two co-axial systems, if every member of one system cuts every member of the other co-axial system orthogonally then the two co-axial systems are called as conjugate co-axial systems to each other.
The length of the common chord of circles x^2 + y^2 - 2x + 1 = 0 and x^2 + y^2 - 5x - 6y + 4 = 0 is
2
22
3
2
The angle at which the circles x^2 + y^2 - 12x - 6y + 41 = 0 and x^2 + y^2 + 4x + 6y - 59 = 0 intersect is
π/2
π/4
π/3
π/2
If the circles x^2 + y^2 - 2kx - 2y - 7 = 0 and 3x^2 + 3y^2 - 8x + 29y = 0 cut orthogonally then k=
-1
1
0
2
The equation of the common chord of the circles x^2 + y^2 - 2x + 1 = 0 and x^2 + y^2 - 5x - 6y + 4 = 0 is
3x + 6y + 5 = 0
x + 2y - 1 = 0
x - 2y - 1 = 0
x + 2y + 1 = 0
The radical axis of the circles 3x^2 + 3y^2 - 2x + 3y + 1 = 0 and x^2 + y^2 - 2y + 1 = 0
2x^2 + 2y^2 - 6x + 5y - 2 = 0
2x - 9y + 2 = 0
10x - 11y + 4 = 0
3x + y = 0
The equation of the circle through the points of intersection of the circles x^2 + y^2 + 4x - 6y - 12 = 0 and x^2 + y^2 + 6x + 4y - 12 = 0 and cutting the circle x^2 + y^2 - 2y + 3 = 0 orthogonally is
x^2 + y^2 + 7x + 9y - 12 = 0
x^2 + y^2 - 7x + 9y - 12 = 0
x^2 + y^2 - 7x - 9y - 12 = 0
x^2 + y^2 - 7x + 9y + 12 = 0
The equation of the circle co-axal with x^2+y^2-6x+4y-8=0 and x^2+y^2+2x+y+4=0 and passing through (0,0) is
3x^2+3y^2+2x+6y=0
3x^2+2y^2-2x-6y=0
3x^2+3y^2-2x+6y=0
3x^2+3y^2+2x-6y=0
The equation of the circle co-axal with x^2+x^2+y^2+8x-6y=0 and x^2+y^2-3x+2y-1=0 and passing through (-1,-2) is
2x^2+2y^2-17x+12y-3=0
2x^2+2y^2-17x+12y+3=0
2x^2+2y^2+17x+12y-3=0
2x^2+2y^2+17x-12y-3=0
The equation of a circle co-axal with x^2+y^2+10x-4y-1=0 and x^2+y^2+5x+y+4=0 are touching the y-axes is
x^2+y^2-8x+2y+1=0
x^2+y^2-6x+9=0
x^2+y^2+8x+2y+1=0
x^2+y^2+8x-2y+1=0
If (1,-1) and (2,0) are the limiting points of a co-axal system then the member of the system which pass through (0,0) is
x^2+y^2+4x=0
x^2+y^2+4x+4y=0
x^2+y^2-4y=0
x^2+y^2+4y=0
If (1,3) and (2,6) are the limiting points of a co-axal system, then the member of the system which pass through (0,0) is
3x^2+3y^2+4x+12y=0
3x^2+3y^2+4x-12y=0
3x^2+3y^2-4x+12y=0
3x^2+3y^2-4x-12y=0
The radical centre of the three circles 1/(x^2+y^2)=1, 0=3x^2+y^2-2x-2y and 0=3x^2+y^2-2y is
(4/3,0)
(-3,0)
(3,0)
(0,-3)
The equation of the circle co-axal with 0=8x^2+y^2-4y and 0=4x^2+y^2+2y and passing through (0,0) is
0=6x^2+3y^2+2y
0=6x^2+3y^2-2y
0=6x^2+3y^2+y
0=6x^2+3y^2-y
If (1,2) is a limiting point of a co-axal system for which the radical axis is 3x+y-10=0 then the other limiting point is
(3,4)
(4,3)
(-3,-4)
(-1,-2)
If (-2,-1) is a limiting point of a co-axal system for which x-y-1=0 is the radical axis, then the other limiting point is
(-4,1)
(0,3)
(0,-3)
(4,-1)
If (1,2) is a limiting point of a co-axal system in which x^2+y^2+2x-6y=0 is a member, then the other limiting point is
(-5,5)
(3,1)
(3,2)
(5,-5)
If (1,3) is a limiting point of a co-axal system in which x^2+y^2-4y=0 is a member, then the other limiting point is
(2,4)
(2,6)
(4,2)
(6,2)
If (3,5) is a limiting point of a co-axal system for which x^2+y^2+2x-24=0 is a member, then the other limiting point is
(1,2)
(-3,-4)
(3,4)
(-1,-2)
A circle passing through the limiting points of a given co-axal system cuts any member of the system at an angle of
45°
90°
60°
30°
If (0,0) is a limiting point of a co-axal system for which S=0 is a member, then the other limiting point is
[gc / (g^2+f^2), fc / g^2+f^2]
[-gc / (g^2+f^2), fc / g^2+f^2]
[-gc / (g^2+f^2), -fc / g^2+f^2]
[gc / (g^2+f^2), -fc / g^2+f^2]
If (0,0) is a limiting point of a co-axal system for which x^2+y^2+2x+2y+4=0 is a member, then the other limiting point is
(1,1)
(-1,-1)
(2,2)
(-2,-2)
The number of common tangents that can be drawn to the circles x^2+y^2-4x+6y+8=0, x^2+y^2-10x-6y+14=0 is
1
2
3
4
The radical centre of the circles x^2+y^2-2x+6y=0,x^2+y^2-4x-2y+6=0,x^2+y^2-12x+12y+30=0 is
(3,1)
(3,0)
(1,3)
(0,3)
The equation of the circle with centre (-1,1) and touching the circle x^2+y^2-4x+6y-3=0 externally is
x^2+y^2+2x-2y+1=0
2x^2+2y^2+12x-2y+1=0
x^2+y^2+2x+12y-11=0
3x^2+3y^2+20x-21y+1=0
The equation of the circle whose radius is 5 and which touches the circle x^2+y^2-2x-4y-20=0 at the point (5,5) is
x^2+y^2-18x-16y+120=0
x^2+y^2+18x+16y-120=0
x^2+y^2-18x-16y-120=0
x^2+y^2+18x+16y+120=0
If the circle x^2+y^2-6x-8y+12=0,x^2+y^2-4x+6y+k=0 cut orthogonally, then k =
-24
24
12
15
If the circles x^2+y^2+2ax+8=0, x^2+y^2+4by-k=0 cut orthogonally, then k=
-24
24
8
-8
The external centre of similitude of the two circles (x-2)^2+(y-1)^2=9, (x+3)^2+(y-1)^2=4 is
(-13,1)
(22,-4)
(2,6)
(6,10)
The equations to the direct common tangents to the circles x^2+y^2+6x+4y+4=0,x^2+y^2-2x=0 is
y-1=0,2x-5y-19=0
y-1=0,-3y-9=0
y+1=0,4x+3y+9=0
y+5=0,2x-2y-25=0
The limiting points of the coaxal system
Common tangents to the circles x^2+y^2+6x+4y+4=0,x^2+y^2-2x=0 is
y-1=0,2x-5y-19=0
y-1=0,-3y-9=0
y+1=0,4x+3y+9=0
y+5=0,2x-2y-25=0
The limiting points of the coaxal system x^2+y^2+11x-5y-2+λ(x-y-1)=0 is
(-2,-1),(0,-3)
(2,1),(-5,-6)
(2,1),(0,3)
(5,2),(5,6)
One limiting point of the coaxal system of circles containing x^2+y^2-6x-6y+4=0 and x^2+y^2-2x-4y+3=0 is
(-1,1)
(-1,2)
(-2,1)
(-2,2)
If a circle passes through the point (a, b) and cuts the circle x^2+y^2=4 orthogonally, then the locus of its centre is
2ax+2by+(a^2+b^2+4)=0
2ax-2by-(a^2+b^2+4)=0
2ax-2by+(a^2+b^2+4)=0
2ax+2by-(a^2+b^2+4)=0
If the two circles (x-1)^2+(y-3)^2=r^2 and x^2+y^2-8x+2y+8=0 intersect in two distinct Points, then
r<2
r=2
r>2
2
The radical centre of the circles x^2+y^2=1,x^2+y^2-2x=1,x^2+y^2-2y=1 is
(0,0)
(1,1)
(1,0)
(0,1)
If (1,2) is a limiting point of a co-axal system for which 3x + y-10 = 0, then the other limiting point is
(3,4)
(4,3)
(-3,-4)
(-1,-2)
If (-2,-1) is a limiting point of a co-axal system for which x –y -1 = 0 is a member, then the other limiting point is
(-4,1)
(0,3)
(0,-3)
(4,-1)
The internal centre of similitude of the two circles x^2+y^2+6x-2y+1=0,x^2+y^2-2x-6y+9=0 is (0,5/2)
only I is true
only II is true
both I and II are true
neither I nor II true
If the equation of the circle of radius 5 and which touches the circle x^2+y^2-2x-4y-20=0 at the point (5,5) is x^2+y^2+2ax+2by+c=0 then the ascending order of a,b,c is
a,b,c
b,c,a
c,a,b
a,c,b
Match the following Circles Property of the circles I. x^2+y^2-4x+6y+8=0,x^2+y^2-10x-6y+14=0 a) touch internally II.x^2+y^2-8x+6y=56=0,x^2+y^2=16 b) touch externally III. x^2+y^2-4x-6y-12=0,x^2+y^2+6x-8y+21=0 c) intersect at two points IV. x^2+y^2-2x-6y+9=0,x^2+y^2+6x-2y+1=0 d) one lies outside the other
a,b,c,d
b,a,c,d
c,a,b,d
d,c,b,a
Match the following. Circles Number of common tangents I. x^2+y^2=4,x^2+y^2-8x+12=0 a)0 II. x^2+y^2=1,x^2+y^2-2x-6y+6=0 b)1 III.x^2+y^2=16,x^2+y^2-8x+6y-56=0 c)2 IV.x^2+y^2-2x-6y+9=0,x^2+y^2+6x-2y+1=0 d)3 e)4
a,b,c,d
d,e,b,a
c,b,e,d
a,c,b,d
Match the following. Circles Radical axis I. x^2+y^2+3x+4y-5=0,x^2+y^2-5x+5y-6=0 a)x+10y-2=0 II. x^2+y^2-6x-4y-44=0,x^2+y^2-14x-5y-24=0 b)8x-y+1=0 III.3x^2+3y^2-7x+8y+11=0,x^2+y^2-3x-4y+5=0 c)8x+y-20=0
a,b,c
b,c,a
c,a,b
a,c,b
The angle of intersection of the circles x^2+y^2=16 and x^2+y^2-2x-4y-4=0
cos^-1(5/6)
20 cos^-1(2/3)
π/4
π/2
The internal centre of similitude of the circles x^2+y^2+6x-2y+1=0, x^2+y^2-2x-6y+1=0 is
(-1,2)
(0,5/2)
(2,0)
(2,5/2)
The external centre of similitude for the circles x^2+y^2-2x-6y+9=0; x^2+y^2=4 is
(2,4)
(2,5)
(5,6)
(2,6)
The circles x^2+y^2-6x-9y+13=0, x^2+y^2-2x-16y=0 touch each other. The common tangent at the point of contact is
4x-7y-13=0
4x-7y+13=0
4x+7y-13=0
4x+7y+13=0
The condition the circles x^2+y^2+2a1x+2b1y=0 & x^2+y^2+2a2x+2b2y=0 touch each other is
a1a2+b1b2=0
a1a2=b1b2
a1b2+a2b1=0
a1b2=a2b1
The condition that the circles x^2+y^2+2ax+c=0 and x^2+y^2+2by+c=0 may touch each other is
c^2=a^2+b^2
c=ab
1/c=1/a+1/b
1/c^2=1/a^2+1/b^2
If the circles x^2+y^2=a^2, x^2+y^2-6x-8y+9=0 touch externally than a=
1
-1
21
16
The equation of the circle passing through (0,0) and cutting orthogonally the circles x^2+y^2+6x-15=0,x^2+y^2-8y+10=0 is
2x^2+2y^2-10x-5y=0
2x^2+2y^2+10x+5y=0
x^2+y^2-5x+5y=0
2x^2+2y^2+10x-5y=0
The distance from (1,2) to the radical axis of the circles x^2+y^2+6x-16=0,x^2+y^2-2x-6y-6=0 is
1
2
√5
√2
The number of common tangents to the circles x^2+y^2+2x+8y-23=0, x^2+y^2-4x-10y+19=0 is
4
2
3
1
The angle at which the circles x^2+y^2+8x-2y-9=0,x^2+y^2-2x+8y-7=0 intersect is
π/6
π/4
π/3
π/2
The angle at which the circles x^2+y^2+8x-2y-9=0,x^2+y^2-2x+8y-7=0 intersect is
π/6
π/4
π/3
π/2
One limiting point of the co-axal system of circles containing x^2+y^2-6x-6y+4=0 and x^2+y^2-2x-4y+3=0 is
(-1,1)
(-1,2)
(-2,1)
(-2,2)
The equation of the circle passing through the origin and the points of intersection of the two circles x^2+y^2-4x-6y-3=0,x^2+y^2+4x-2y-4=0 is
x^2+y^2-28x-18y=0
x^2+y^2+28x-18y=0
x^2+y^2+28x+18y=0
x^2+y^2-28x+18y=0
The number of common tangents to the two circles x^2+y^2=4,x^2+y^2-8x+12=0 is
1
2
3
4
The distance of the point (1,2) from the common chord of the circles x^2+y^2-2x+3y-5=0 and x^2+y^2+10x+8y-1=0 is
2
1
√2
√3
The circles x^2+y^2-4x+6y+8=0, x^2+y^2-10x-6y+14=0
touch internally
touch externally
intersecting at two points
are such that one completely lies outside the other
The equation of the circle described on the common chord of the circles x^2+y^2+2x=0, x^2+y^2+2y=0 as diameter is
x^2+y^2+x-y=0
x^2+y^2-x-y=0
x^2+y^2-x-y=0
x^2+y^2+x+y=0
Consider the circles x^2+(y-1)^2=9,(x-1)^2+y^2=25. They are such that
these circles touch each other
one of these circles lies entirely inside the other
each of these circles lies outside the other
they intersect in two points
A circle passes through origin and has its centre on y=x. If it cuts x^2+y^2-4x-6y+10=0 orthogonally then the equation of the circle is
x^2+y^2-x-y=0
x^2+y^2-4x-4y=0
x^2+y^2-6x-3y=0
x^2+y^2+2x+2y=0
The number of common tangents to the two circles x^2+y^2-x=0,x^2+y^2+x=0 is
2
1
3
3
Origin is a limiting point of a coaxal system of which x^2+y^2-6x-8y+1=0 is a member. The other limiting point is
(-2,-4)
(3/25,4/25)
(-3/25,-4/25)
(4/25,3/25)
A circle of the co-axal system with limiting points (0,0) and (1,0) is
x^2+y^2-3x=0
x^2+y^2-6x+3=0
x^2+y^2=1
x^2+y^2-x+1=0
The distance of the point (1,-2) from the common chord of the circles x^2+y^2-5x+4y-2=0 and x^2+y^2-2x+8y+3=0 is
0
1
2
3
The radical centre of the circles x^2+y^2-x+3y-3=0,x^2+y^2-2x+2y+2=0,x^2+y^2+2x+3y-9=0 is
(2,3)
(2,-3)
(-2,3)
(-2,-3)
If (0,0) is one limiting point of a co-axal system of circles whose common radical axis is the line x+y=1 then the other limiting point is
(1,1)
(-1,-1)
(1,-1)
(-1,1)
The radical axis of the co-axal system of circles with limiting points (1,2) and (4,3) is given by the equation
3x-y+10=0
3x+y-10=0
x-y-1=0
4x-2y-5=0
The radical axis of the circles x^2+y^2-6x-4y-44=0 and x^2+y^2-14x-5y-24=0 is
8x+y-30=0
8x+y+20=0
8x+3y-20=0
8x+y-20=0
The slope of the radical axis of the circles x^2+y^2+3x+4y-5=0 and x^2+y^2-5x+5y-6=0 is
1
3
5
8
If the circle x^2+y^2+2x-2y+4=0 cuts the circle x^2+y^2+4x-2fy+2=0 Orthogonally, then f=
1
2
-1
-2
If the circles of same radius a and centres (2,3),(5,6) cut orthogonally, then a=
1
2
3
4
If (1,2) is a limiting point of the co-axal system of circles containing the circle x^2+y^2+x-5y+9=0 then the equation of the radical axes is
x+3y+9=0
3x-y+4=0
x+9y-4=0
3x-y-1=0
If the circles of same radius a and centres (2,3),(5,6) cut orthogonally, then a=
1
2
3
4
The number of common tangents that can be drawn to the circles x^2+y^2=1 and x^2+y^2-2x-6y+6=0 is
1
2
3
4
The radical axis of the circles x^2+y^2+3x+4y-5=0 and x^2+y^2-5x+5y-6=0 is
8y-x+1=0
8x-y+1=0
8x-8y+1=0
y-8x+1=0
The limiting points of the co-axal system containing the two circles x^2+y^2+2x-2y+2=0 and 25(x^2+y^2)-10x-80y+65=0 are
(1,-1),(-5,-40)
(1,-1),(-1/5,-8/5)
The points of the coaxial system containing the two circles x^2+y^2+2x-2y+2=0 and 25(x^2+y^2)-10x-80y+65=0 are
(1,-1),(-5,-40)
(1,-1),(-1/5,-8/5)
(-1,1),(1/5,8/5)
(-1,1),(-1/5,-8/5)
A line l meets the circle x^2+y^2=61 in A,B and p(-5,6) is such that PA=PB=10. Then the equation of l is
5x+6y+11=0
5x-6y-11=0
5x-6y+11=0
5x-6y+122=0
If x-y+1=0 meets the circle x^2+y^2+y-1=0 at A and B, then the equation of the circle with AB as diameter is:
2(x^2+y^2)+3x-y+1=0
2(x^2+y^2)+3x-y+2=0
2(x^2+y^2)+3x-y+3=0
x^2+y^2+3x-y+1=0
The number of common tangents to the two circles x^2+y^2-8x+2y=0 and x^2+y^2-2x-16y+25=0 is:
1
2
3
4
Observe the following statements: I. The circle x^2+y^2-6x-4y-7=0 touches y-axis II. The circle x^2+y^2+6x+4y-7=0 touches x-axis Which of the following is a correct statement?
Both I and II are true
Neither I nor II is true
I is true, II is false
I is false, II is true
The length of tangent drawn to the circle x^2+y^2-2x+4y-11=0 from the point (1,3) is:
1
2
3
4
The condition for the coaxial system x^2+y^2+2λx +c = 0, where λ is a parameter and c is a constant, to have distinct limiting points is
c = 0
c < 0
c =-1
c > 0
If θ is the angle between the tangents from (-1,0) to the circle x^2+y^2-4x-6y+9=0 is
(1,1/2)
(2,1)
(0,1)
(1,0)
The point (3, –4) lies on both the circles x^2+ y^2– 2x + 8y + 13 = 0 and x^2+ y^2– 4x + 6y + 11 = 0. Then the angle between the circles is:
60
tan^-1(2/1)
tan^-1(5/3)
135
The equation of the circle which passes through the origin and cuts orthogonally each of the circles x^2+y^2–6x+8 = 0 and x^2+y^2–2x–2y = 7 is:
3x^2+3y^2–8x–13y = 0
3x^2+3y^2–8x–29y = 0
3x^2+3y^2+8x+29y = 0
3x^2+3y^2–8x+29y = 0
If the circle 22x^2 + 3y^2 + 1 + 0 = 0 cuts another circle 22x^2 + 4y^2 + 3 + 0 = 0 in A and B, then the equations of the circle with AB as a diameter is:
22x^2 + 3y^2 + 0 = 0
22x^2 + 2y^2 + 2 + 6 + 1 = 0
22x^2 + 6y^2 + 1 + 0 = 0
22x^2 + 2y^2 + 23 + 1 = 0
The length of the common chord of the circles of radii 15 and 20 whose centers are 25 units of distance apart, is:
12
16
24
25
If the circle x^2 + y^2 + 8x – 4y + c = 0 touches the circle x^2 + y^2 + 2x + 4y – 11 = 0 externally and cuts the circle x^2 + y^2 – 6x + 8y + k = 0 orthogonally then k =
59
-59
19
-19
The point of contact of the circle x^2+y^2+2x+2y+1=0 and x^2+y^2–2x+2y+1=0 is
(0, 1)
(0, -1)
(1, 0)
(-1, 0)
The equation to the line joining the centres of the circles belonging to the coaxial system of circles 4x^2 + 4y^2 – 12x + 6y – 3 + (x + 2y – 6) = 0 is
8x – 4y – 15 = 0
8x – 4y + 15 = 0
3x – 4y – 5 = 0
3x – 4y + 5 = 0
(a, 0) & (b, 0) are centres of two circles belonging to a coaxial system of which y – axis is the radical axis. If radius of one of the circles is ‘r’, then the radius of the other circle is
(r^2 + b^2 + a^2)^(1/2)
(r^2 + b^2 - a^2)^(1/2)
(r^2 + b^2 - a^2)^(1/3)
(r^2 + b^2 + a^2)^(1/3)
If the circle x^2+y^2+4x–6y+c=0 bisects the circumference of the circle x^2+y^2–6x+4y–12=0, then c =
16
24
-42
-62
The length of the common chord of the two circles x^2+y^2–4y=0 & x^2+y^2–8x–y+11=0 is
The locus of the centre of the circle which cuts the circle x^2+y^2–20x+4=0 orthogonally & touches the lune x = 2 is
y^2=4x
y^2=16x
x^2=4y
x^2=16y
The length of the common chord of the two circles (x–a)^2+y^2=a^2 & x^2+(y–b)^2=b^2 is
The equation of the circles passing through (1, 2) & the points of intersection of the circles x^2+y^2–8x–6y+21=0 & x^2+y^2–2x–15=0
x^2+y^2+6x–2y+9=0
x^2+y^2–6x–2y+9=0
x^2+y^2–6x–4y+9=0
x^2+y^2–6x+4y+9=0
