IISER Mathematics - Circle
Exam Duration: 45 Mins Total Questions : 30
Equation of the diameter of the circle \({ x }^{ 2 }+{ y }^{ 2 }-12x+4y+6=0\) is given by
- (a)
x + y = 0
- (b)
x + 3y = 0
- (c)
x = y
- (d)
3x + 2y = 0
The equation of the circle which passes through the origin and makes intercept of lengths a and b on the X and Y-axes respectively, are
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }\pm ax\pm by=0\)
- (b)
\({ x }^{ 2 }\pm { y }^{ 2 }=0\)
- (c)
\({ x }^{ 2 }\pm ax\pm by=0\)
- (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
Find the equation of a circle concentric with the circle \({ X }^{ 2 }+{ Y }^{ 2 }-6X+12Y+15=0\) and has double of its area.
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }-6x+12y-15=0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }-6x-12y+15=0\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }-6x+12y+15=0\)
- (d)
None of the above
Let \({ X }^{ 2 }+{ Y }^{ 2 }+2gX+2fY+c=0\) be an equation of circle.
Column I | Column II |
A. If circle lies in first quadrant, then | p. g < 0 |
B. If circle lies above X-axis, then | q. g > 0 |
C. If circle lies on the left of Y-axis, then | r. g2 - c < 0 |
D. If circle touches positive X-axis and does not intersect Y.axis then | s. c > 0 |
- (a)
A B C D prs rs qs ps - (b)
A B C D pqr sq pr ps - (c)
A B C D pr ps pq qrs - (d)
None of the above
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 point diametrically opposite to the point P(1,0) on the circle \({ X }^{ 2 }+{ Y }^{ 2 }+2X+4Y-3=0\) is
- (a)
(3, 4)
- (b)
(3, -4)
- (c)
(-3, 4)
- (d)
(-3, -4)
If a circle passes through the point (a,b) and cuts the circle \({ X }^{ 2 }+{ Y }^{ 2 }={ p }^{ 2 }\) orthogonally, then the equation of the locus of its centre is
- (a)
\(2ax+2by-\left( { a }^{ 2 }+{ b }^{ 2 }+{ p }^{ 2 } \right) =0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }-2ax-3by+\left( { a }^{ 2 }-{ b }^{ 2 }-{ p }^{ 2 } \right) =0\)
- (c)
\(2ax+2by-\left( { a }^{ 2 }+{ b }^{ 2 }+2{ b }^{ 2 }+{ p }^{ 2 } \right) =0\)
- (d)
\({ x }^{ 2 }+{ y }^{ 2 }-3ax-4by+\left( { a }^{ 2 }+{ b }^{ 2 }-{ p }^{ 2 } \right) =0\)
The number of points (x, y) having integral coordinates satisfying the condition x2+ y2 < 25 is
- (a)
69
- (b)
80
- (c)
81
- (d)
77
The number of rational point(s) (a point (a, b) is rational, if a and b bath are rational numbers) on the circumference of a circle having centre (ㅠ,e) is
- (a)
at most one
- (b)
at least two
- (c)
exactly two
- (d)
infinite
AB is a diameter of a circle and C is any point on the circumference of the circle. Then
- (a)
the area of ∆ABC is maximum when it is isosceles
- (b)
the area of ∆ABC is minimum when it is isosceles
- (c)
the perimeter of ∆ABC is maximum when it is isosceles
- (d)
none of the above
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
If one circle of a coaxial system is x2+y2+2gx+2fy+c=0 and one limiting point is (a, b), then equation of the radical axis will be
- (a)
(g + a)x + (f + b)y + c - a2 - b2 = 0
- (b)
2(g+a)x+2(f+b)y+c-a2-b2=0
- (c)
2gx+2fy+c-a2-b2=0
- (d)
none of the above
The centre of the circle r2=2-4r cosθ+6r sinθ is
- (a)
(2,3)
- (b)
(-2,3)
- (c)
(-2,-3)
- (d)
(2,-3)
If two circles (x-1)2+(y-3)2=r2 and x2+y2-8x+2y+8=0 intersect in two distinct points, then
- (a)
2<r<8
- (b)
r < 2
- (c)
r=2
- (d)
r > 2
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
∝, β and ૪ are parametric angles of three points P, Q and R respectively, on the circle x2 + y2 = 1and A is the point (-1, 0). If the lengths of the chords AP, AQ and AR are in GP, then cos ∝/2, cos β/2 and cos ૪/2 are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
The equation of the circle passing through (2, 0) and (0,4) and having the minimum radius is
- (a)
x2 + y2 = 20
- (b)
x2 + y2 - 2x - 4y = 0
- (c)
(x2 + y2 - 4) + A(x2 + y2 -16) = 0
- (d)
none of the above
A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is
- (a)
m(m + n)
- (b)
(m + n)
- (c)
n(m + n)
- (d)
\(\frac{1}{2}\)(m+n)
The tangents drawn from the origin to the circle x2 + y2 - 2px - 2qy + q2 = 0 are perpendicular, if
- (a)
p=q
- (b)
p2=q2
- (c)
q=-p
- (d)
p2+q2=1
If the circle x2 + y2 + 2gx + 2fy + c = 0 cuts each of the circles x2+y2-4=0, x2+y2-6x-8y+10=0 and x2 +y2 + 2x - 4y - 2 = 0 at the extremities of a diameter, then
- (a)
c=-4
- (b)
g + f = c - 1
- (c)
g2+f2-c=17
- (d)
gf=6
Equation of the circle having diameters x - 2y + 3 = 0, 4x - 3y + 2 = 0 and radius equal to 1 is
- (a)
(x - 1)2 + (y - 2)2 = 1
- (b)
(x - 2)2 + (y - 1)2 = 1
- (c)
x2 + y2 - 2x - 4y + 4 = 0
- (d)
x2 + y2 - 3x - 4y + 7 = 0
The equation of a circle is S1≡x2+y2=1. The orthogonal tangents to S1 meet at another circle S2 and the orthogonal tangents to S2 meet at the third circle S3 Then
- (a)
radius of S2 and S3 are in the ratio 1:√2
- (b)
radius of S2 and S3 are in the ratio 1 : 2
- (c)
the circles S1 , S2 and S3 are concentric
- (d)
none of the above
If (a cosθi ,a sinθi); i=1,2,3 represent the vertices of an equilateral triangle inscribed in a circle, then
- (a)
cos θ1+cos θ2+cos θ3=0
- (b)
sin θ1+sin θ2+sin θ3=0
- (c)
tan θ1+tan θ2+tan θ3=0
- (d)
cot θ1+cot θ2+cot θ3=0
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 |n|≠ 1, then the locus of a point P is
- (a)
a straight line
- (b)
a circle
- (c)
a parabola
- (d)
an ellipse
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
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 n > 1, then
- (a)
A lies inside the circle 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 16m2-8l-1=0 then equation of the circle having Ix + my + 1=0 is a tangent is
- (a)
x2+y2+8x=0
- (b)
x2+y2-8x=0
- (c)
x2+y2+8y=0
- (d)
x2+y2-8y=0
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 16l2+9m2=24lm+6l+8m+1 and if S be the equation of the circle having lx+my+1=0 is a tangent when the equation of director circle of S is
- (a)
x2+y2+6x+8y=25
- (b)
x2+y2-6x+8y=25
- (c)
x2+y2-6x-8y=25
- (d)
x2+y2+6x-8y=25
P is a variable point on the line L = 0.Tangents are drawn to the circle x2 + y2 = 4 from P to touch it at Q and R. The parallelogram PQSR is completed.
If P≡(2,3) then the centre of circumcircle of triangle QRS is
- (a)
\(\left( \frac { 2 }{ 13 } ,\frac { 7 }{ 26 } \right) \)
- (b)
\(\left( \frac { 2 }{ 13 } ,\frac { 3 }{ 26 } \right) \)
- (c)
\(\left( \frac { 3 }{ 13 } ,\frac { 9 }{ 26 } \right) \)
- (d)
\(\left( \frac { 3 }{ 13 } ,\frac { 2 }{ 13 } \right) \)