JEE Main Mathematics - Circle
Exam Duration: 60 Mins Total Questions : 30
Find the equation of a circle which touches both the axes and the line 3X - 4Y + 8 = 0 and lies in the third quadrant.
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }+4x+4y-4=0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }-4x-4y+4=0\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }+4x+4y+4=0\)
- (d)
None of the above
\({ X }^{ 2 }+{ Y }^{ 2 }+4X-6Y+4=0\) is the equation of the incircle of an equilateral triangle, then the equation of the circumcircle of the triangle is
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }+4x+6y-23=0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }+4x-6y-23=0\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }-4x-6y-23=0\)
- (d)
None of the above
The circle \({ x }^{ 2 }+{ y }^{ 2 }={ r }^{ 2 }\) intersects the line Y = mX + c at the two real distinct points, if
- (a)
\(-r\sqrt { 1+{ m }^{ 2 } }\)
- (b)
\(-c\sqrt { 1-{ m }^{ 2 } }\)
- (c)
\(-r\sqrt { 1-{ m }^{ 2 } }\)
- (d)
None of the above
If a circle passes through the point (0,0), (a,0) and (0,b), then find the coordinates of its centre.
- (a)
\(\left( -\frac { a }{ 2 } ,-\frac { b }{ 2 } \right) \)
- (b)
\(\left( \frac { a }{ 2 } ,-\frac { b }{ 2 } \right) \)
- (c)
\(\left( -\frac { a }{ 2 } ,\frac { b }{ 2 } \right) \)
- (d)
None of these
The equation of the smallest circle passing through the intersection of line x + Y = 2 and the circle \({ X }^{ 2 }+{ Y }^{ 2 }=16\) is
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }-2x-2y-12=0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }-2x+2y-12=0\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }+2x+2y+12=0\)
- (d)
\({ x }^{ 2 }+{ y }^{ 2 }+2x-2y-12=0\)
The circle \({ X }^{ 2 }+{ Y }^{ 2 }=4X+8Y+5\) intersects the line \(3X-4Y=m\) at two distinct points, if
- (a)
\(-85
- (b)
\(-35
- (c)
\(15
- (d)
\(35
The range of values of ,a' such that the angle θ between the pair of tangents drawn from (a, 0) to the circle x2+y2=1 satisfies \(\frac { \pi }{ 2 } <\theta <\pi \)is
- (a)
(1,2)
- (b)
(1,√2)
- (c)
(-√2,-1)
- (d)
(-√2,-1)U(1,√2)
The centres of a set of circles, each of radius 3, lie on the circle x2 + y2 = 25. The locus of any point in the set is
- (a)
4\(\le\)x2+y2\(\le\)64
- (b)
x2+y2\(\le\)25
- (c)
x2+y2\(\ge\)25
- (d)
3\(\le\)x2+y2\(\le\)9
If a chord of the circle x2+y2=8 makes equal intercepts of length a on the coordinate axes, then
- (a)
\(\left| a \right| \)<8
- (b)
\(\left| a \right| \)<4\(\sqrt{2}\)
- (c)
\(\left| a \right| \)<4
- (d)
\(\left| a \right| \)>4
One of the diameter of the circle x2+y2-12x+4y+6=0 is given by
- (a)
x+y=0
- (b)
x+3y=0
- (c)
x=y
- (d)
3x+2y=0
The locus of the point of intersection of the tangents to the circle x=r cosθ, y=r sinθ at points whose parametric angles differ by ㅠ/3 is
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }=4(2-\sqrt { 3 } ){ r }^{ 2 }\)
- (b)
3(x2+y2)=1
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }=(2-\sqrt { 3 } ){ r }^{ 2 }\)
- (d)
3(x2+y2)=4r2
If the distances from the origin of the centres of three circles \({ x }^{ 2 }+{ y }^{ 2 }+2{ \lambda }_{ i }x-{ c }^{ 2 }\) = 0 (i = 1, 2, 3) are in GP, then the lengths of the tangents drawn to them from any point on the circle x2 + y2 = c2 are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
The circles (x - a)2 + (y - b)2 = c2 and (x - b)2 + (y - a)2 = c2 touch each other, then
- (a)
a = b ± 2c
- (b)
a = b ±√2c
- (c)
a = b ± c
- (d)
none of these
The locus of the point of intersection of the lines \(x=a\left( \frac { 1-{ t }^{ 2 } }{ 1+{ t }^{ 2 } } \right) \) and y=\(\frac { 2at }{ 1+{ t }^{ 2 } } \)represents (t being a parameter)
- (a)
circle
- (b)
parabola
- (c)
ellipse
- (d)
hyperbola
If f(x + y) = f(x)· f(y) for all x and y, f(1) = 2 and ∝n=f(n), n∈N, then the equation of the circle having (∝1, ∝2) and (∝3,∝4) as the ends of its one diameter is
- (a)
(x - 2)(x - 8) + (y - 4)(y -16) = 0
- (b)
(x - 4)(x - 8) + (y - 2)(y - 16) = 0
- (c)
(x - 2)(x -16) + (y - 4)(y - 8) = 0
- (d)
(x-6)(x-8)+(y-5)(y-6)=0
If P is a point on the circle x2 + y2 = 9 Q is a point on the line 7x + y + 3 = 0, and the line x - y + 1 = 0, is the perpendicular bisector of PQ, then the coordinates of Pare
- (a)
(3,0)
- (b)
\(\left( \frac { 72 }{ 25 } ,-\frac { 21 }{ 25 } \right) \)
- (c)
(0,3)
- (d)
\(\left( -\frac { 72 }{ 25 } ,\frac { 21 }{ 25 } \right) \)
The equation of a circle C1 is x2 + y2 = 4. The locus of the intersection of orthogonal tangents to the circle is the curve C2 and the locus of the intersection of perpendicular tangents to the curve C2 is the curve C3 Then
- (a)
C3 is a circle
- (b)
the area enclosed by the curve C3 is 8ㅠ
- (c)
C2 and C3 are circles with the same centre
- (d)
none of the above
S≡x2+y2+2x+3y+1=0 and S'≡x2+y2+4x+3y+2=0 are two circles. The point (-3, - 2) lies
- (a)
inside S' only
- (b)
inside S only
- (c)
inside S and S'
- (d)
outside S and S'
The locus of a point which moves such that the tangents from it to the two circles x2 + y2 - 5x - 3 = 0 and 3x2 + 3y2 + 2x + 4y - 6 = 0 are equal, is
- (a)
2x2 + 2y2 + 7x + 4y - 3 = 0
- (b)
17x+4y+3=0
- (c)
4x2 + 4y2 - 3x + 4y - 9 = 0
- (d)
13x - 4y + 15 = 0
The circles x2 + y2 - 4x - 81 = 0, x2 + y2 + 24x - 81 = 0 intersect each other at points A and B. A line through point A meet one circle at P and a parallel line through B meet the other circle at Q. Then the locus of the mid point of PQ is
- (a)
(x + 5)2 + (y + 0)2 = 25
- (b)
(x - 5)2 + (y - 0)2 = 25
- (c)
x2+y2+10x=0
- (d)
x2+y2-10x=0
If the line 3x - 4y -⋋=0 touches the circle x2+y2-4x - 8y - 5 = 0 at (a, b), then ⋋+a+b is equal to
- (a)
20
- (b)
22
- (c)
-30
- (d)
-28
Let A ≡ (a, 0) and B ≡ (-a, 0) be two fixed points ∀ a ∈ (-∞, 0) and P moves on a plane such that PA = nPB (n≠ 0).
If 0
- (a)
A lies inside the circle and B lies outside the circle
- (b)
A lies outside the circle and B lies inside the circle
- (c)
both A and B lies on the circle
- (d)
both A and B lies inside the circle
If 7l2 - 9m2 + 81 + 1 = 0 and we have to find equation of circle having Ix + my + 1 = 0 is a tangent and we can adjust given condition as 16l2 + 8l + 1 = 9 (l2 + m2)
or (4l+1)2=9(l2+m2)⇒ \(\frac { \left| 4l+1 \right| }{ \sqrt { \left( { l }^{ 2 }+{ m }^{ 2 } \right) } } =3\)
Centre of circle = (4, 0) and radius = 3 when any two non parallel lines touching a circle, then centre of circle lies on angle bisector of lines.
If 4l2-5m2 +6l+1=0, then the centre and radius of the circle which have lx+my+1=0 is a tangent is
- (a)
(0,4);√5
- (b)
(4,0);√5
- (c)
(0,3);√5
- (d)
(3,0);√5