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MCQ (Single Correct Answer)

JEE Main 2024 (Online) 9th April Morning Shift
Let a circle passing through $$(2,0)$$ have its centre at the point $$(\mathrm{h}, \mathrm{k})$$. Let $$(x_{\mathrm{c}}, y_{\mathrm{c}})$$ be the poin...
JEE Main 2024 (Online) 4th April Morning Shift
A square is inscribed in the circle $$x^2+y^2-10 x-6 y+30=0$$. One side of this square is parallel to $$y=x+3$$. If $$\left(x_i, y_i\right)$$ are the ...
JEE Main 2024 (Online) 1st February Evening Shift
Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $\mathrm{P}$ and $\math...
JEE Main 2024 (Online) 1st February Morning Shift
Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ an...
JEE Main 2024 (Online) 31st January Evening Shift
Let a variable line passing through the centre of the circle $$x^2+y^2-16 x-4 y=0$$, meet the positive co-ordinate axes at the points $$A$$ and $$B$$....
JEE Main 2024 (Online) 31st January Morning Shift
If one of the diameters of the circle $$x^2+y^2-10 x+4 y+13=0$$ is a chord of another circle $$\mathrm{C}$$, whose center is the point of intersection...
JEE Main 2024 (Online) 30th January Morning Shift
If the circles $$(x+1)^2+(y+2)^2=r^2$$ and $$x^2+y^2-4 x-4 y+4=0$$ intersect at exactly two distinct points, then
JEE Main 2024 (Online) 27th January Morning Shift
Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :
JEE Main 2023 (Online) 15th April Morning Shift
The number of common tangents, to the circles $x^{2}+y^{2}-18 x-15 y+131=0$ and $x^{2}+y^{2}-6 x-6 y-7=0$, is :
JEE Main 2023 (Online) 13th April Evening Shift
Let the centre of a circle C be $$(\alpha, \beta)$$ and its radius $$r ...
JEE Main 2023 (Online) 10th April Evening Shift
Let A be the point $$(1,2)$$ and B be any point on the curve $$x^{2}+y^{2}=16$$. If the centre of the locus of the point P, which divides the line seg...
JEE Main 2023 (Online) 10th April Morning Shift
A line segment AB of length $$\lambda$$ moves such that the points A and B remain on the periphery of a circle of radius $$\lambda$$. Then the locus o...
JEE Main 2023 (Online) 8th April Evening Shift
Let O be the origin and OP and OQ be the tangents to the circle $$x^2+y^2-6x+4y+8=0$$ at the points P and Q on it. If the circumcircle of the triangle...
JEE Main 2023 (Online) 6th April Evening Shift
If the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the circle $$x^{2}+y^{2}-2 x+y=5$$ meet at the point $$R\left(\frac{9}{4}, 2\right)...
JEE Main 2023 (Online) 31st January Evening Shift
The set of all values of $a^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{...
JEE Main 2023 (Online) 31st January Morning Shift
Let a circle $$C_{1}$$ be obtained on rolling the circle $$x^{2}+y^{2}-4 x-6 y+11=0$$ upwards 4 units on the tangent $$\mathrm{T}$$ to it at the point...
JEE Main 2023 (Online) 30th January Morning Shift
Let $$y=x+2,4y=3x+6$$ and $$3y=4x+1$$ be three tangent lines to the circle $$(x-h)^2+(y-k)^2=r^2$$. Then $$h+k$$ is equal to :
JEE Main 2023 (Online) 29th January Morning Shift
Let the tangents at the points $$A(4,-11)$$ and $$B(8,-5)$$ on the circle $$x^{2}+y^{2}-3 x+10 y-15=0$$, intersect at the point $$C$$. Then the radius...
JEE Main 2023 (Online) 25th January Morning Shift
The points of intersection of the line $$ax + by = 0,(a \ne b)$$ and the circle $${x^2} + {y^2} - 2x = 0$$ are $$A(\alpha ,0)$$ and $$B(1,\beta )$$. T...
JEE Main 2023 (Online) 24th January Evening Shift
The locus of the mid points of the chords of the circle $${C_1}:{(x - 4)^2} + {(y - 5)^2} = 4$$ which subtend an angle $${\theta _i}$$ at the centre o...
JEE Main 2022 (Online) 28th July Evening Shift
Let the tangents at two points $$\mathrm{A}$$ and $$\mathrm{B}$$ on the circle $$x^{2}+\mathrm{y}^{2}-4 x+3=0$$ meet at origin $$\mathrm{O}(0,0)$$. Th...
JEE Main 2022 (Online) 28th July Morning Shift
For $$\mathrm{t} \in(0,2 \pi)$$, if $$\mathrm{ABC}$$ is an equilateral triangle with vertices $$\mathrm{A}(\sin t,-\cos \mathrm{t}), \mathrm{B}(\opera...
JEE Main 2022 (Online) 28th July Morning Shift
Let $$C$$ be the centre of the circle $$x^{2}+y^{2}-x+2 y=\frac{11}{4}$$ and $$P$$ be a point on the circle. A line passes through the point $$\mathrm...
JEE Main 2022 (Online) 27th July Evening Shift
A circle $$C_{1}$$ passes through the origin $$\mathrm{O}$$ and has diameter 4 on the positive $$x$$-axis. The line $$y=2 x$$ gives a chord $$\mathrm{...
JEE Main 2022 (Online) 27th July Morning Shift
If the circle $$x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}$$ passes through the point $$(6,1)$$ and its centre lies on the line $$x-2 c y=8$$, ...
JEE Main 2022 (Online) 26th July Evening Shift
Let the abscissae of the two points $$P$$ and $$Q$$ on a circle be the roots of $$x^{2}-4 x-6=0$$ and the ordinates of $$\mathrm{P}$$ and $$\mathrm{Q}...
JEE Main 2022 (Online) 30th June Morning Shift
Consider three circles: $${C_1}:{x^2} + {y^2} = {r^2}$$ $${C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}$$ $${C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}$$ If a l...
JEE Main 2022 (Online) 29th June Evening Shift
Let a triangle ABC be inscribed in the circle $${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$ such that $$\angle BAC = {\pi \over 2}$$. If the length of side...
JEE Main 2022 (Online) 29th June Morning Shift
Let the tangent to the circle C1 : x2 + y2 = 2 at the point M($$-$$1, 1) intersect the circle C2 : (x $$-$$ 3)2 + (y $$-$$ 2)2 = 5, at two distinct po...
JEE Main 2022 (Online) 28th June Morning Shift
If the tangents drawn at the points $$O(0,0)$$ and $$P\left( {1 + \sqrt 5 ,2} \right)$$ on the circle $${x^2} + {y^2} - 2x - 4y = 0$$ intersect at the...
JEE Main 2022 (Online) 27th June Evening Shift
The set of values of k, for which the circle $$C:4{x^2} + 4{y^2} - 12x + 8y + k = 0$$ lies inside the fourth quadrant and the point $$\left( {1, - {1 ...
JEE Main 2022 (Online) 26th June Morning Shift
Let C be a circle passing through the points A(2, $$-$$1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its cent...
JEE Main 2022 (Online) 25th June Evening Shift
A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :
JEE Main 2022 (Online) 25th June Morning Shift
Let a circle C touch the lines $${L_1}:4x - 3y + {K_1} = 0$$ and $${L_2} = 4x - 3y + {K_2} = 0$$, $${K_1},{K_2} \in R$$. If a line passing through the...
JEE Main 2021 (Online) 27th August Evening Shift
Let Z be the set of all integers,$$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $$$$B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $$$$C...
JEE Main 2021 (Online) 26th August Evening Shift
A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C1 : x2 + y2 + 2y $$-$$ 5 = 0 at two points P and Q such that PQ is a...
JEE Main 2021 (Online) 26th August Morning Shift
If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point ($$-$$30, 0) and is tangent to the parabola y2 = 30x, then ...
JEE Main 2021 (Online) 27th July Evening Shift
Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $$6\sqrt 5 $$ on the x-axis. Then the radius of the circle C is equal...
JEE Main 2021 (Online) 27th July Morning Shift
Two tangents are drawn from the point P($$-$$1, 1) to the circle x2 + y2 $$-$$ 2x $$-$$ 6y + 6 = 0. If these tangents touch the circle at points A and...
JEE Main 2021 (Online) 27th July Morning Shift
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line ...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\} $$, $$B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\} $$ and $$C = \{ (x...
JEE Main 2021 (Online) 22th July Evening Shift
Let the circle S : 36x2 + 36y2 $$-$$ 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersec...
JEE Main 2021 (Online) 20th July Evening Shift
Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point ($$-$$4, 1) and having their centres on the...
JEE Main 2021 (Online) 18th March Evening Shift
Let S1 : x2 + y2 = 9 and S2 : (x $$-$$ 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally alw...
JEE Main 2021 (Online) 18th March Morning Shift
Choose the correct statement about two circles whose equations are given below :x2 + y2 $$-$$ 10x $$-$$ 10y + 41 = 0x2 + y2 $$-$$ 22x $$-$$ 10y + 137 ...
JEE Main 2021 (Online) 18th March Morning Shift
For the four circles M, N, O and P, following four equations are given :Circle M : x2 + y2 = 1Circle N : x2 + y2 $$-$$ 2x = 0Circle O : x2 + y2 $$-$$ ...
JEE Main 2021 (Online) 17th March Evening Shift
Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circ...
JEE Main 2021 (Online) 17th March Evening Shift
Two tangents are drawn from a point P to the circle x2 + y2 $$-$$ 2x $$-$$ 4y + 4 = 0, such that the angle between these tangents is $${\tan ^{ - 1}}\...
JEE Main 2021 (Online) 17th March Morning Shift
The line 2x $$-$$ y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x $$-$$ 2y = 4. Then, the radius of th...
JEE Main 2021 (Online) 17th March Morning Shift
Choose the incorrect statement about the two circles whose equations are given below :x2 + y2 $$-$$ 10x $$-$$ 10y + 41 = 0 and x2 + y2 $$-$$ 16x $$-$$...
JEE Main 2021 (Online) 16th March Evening Shift
Let the lengths of intercepts on x-axis and y-axis made by the circle x2 + y2 + ax + 2ay + c = 0, (a < 0) be 2$${\sqrt 2 }$$ and 2$${\sqrt 5 }$$, r...
JEE Main 2021 (Online) 26th February Evening Shift
Let A(1, 4) and B(1, $$-$$5) be two points. Let P be a point on the circle (x $$-$$ 1)2 + (y $$-$$ 1)2 = 1 such that (PA)2 + (PB)2 have maximum value,...
JEE Main 2021 (Online) 26th February Evening Shift
If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then r is equal...
JEE Main 2021 (Online) 26th February Morning Shift
In the circle given below, let OA = 1 unit, OB = 13 unit and PQ $$ \bot $$ OB. Then, the area of the triangle PQB (in square units) is :...
JEE Main 2020 (Online) 5th September Evening Slot
If the length of the chord of the circle, x2 + y2 = r2 (r > 0) along the line, y – 2x = 3 is r, then r2 is equal to : ...
JEE Main 2020 (Online) 4th September Evening Slot
The circle passing through the intersection of the circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having its centre on the line, 2x – 3y + 12 = 0, al...
JEE Main 2020 (Online) 9th January Morning Slot
A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?
JEE Main 2020 (Online) 8th January Evening Slot
If a line, y = mx + c is a tangent to the circle, (x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle, x2 + ...
JEE Main 2020 (Online) 7th January Evening Slot
Let the tangents drawn from the origin to the circle, x2 + y2 - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)2 is equal to :...
JEE Main 2019 (Online) 12th April Evening Slot
A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :
JEE Main 2019 (Online) 12th April Morning Slot
If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90o, then the length (in cm) of their common chor...
JEE Main 2019 (Online) 10th April Evening Slot
The locus of the centres of the circles, which touch the circle, x2 + y2 = 1 externally, also touch the y-axis and lie in the first quadrant, is :...
JEE Main 2019 (Online) 10th April Morning Slot
If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y –1 = 0, (K$$ \in $$R), intersect at the points P and Q, then the line 4x + 5y –...
JEE Main 2019 (Online) 10th April Morning Slot
The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its radius is :
JEE Main 2019 (Online) 9th April Evening Slot
The common tangent to the circles x 2 + y2 = 4 and x2 + y2 + 6x + 8y – 24 = 0 also passes through the point :...
JEE Main 2019 (Online) 9th April Evening Slot
A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6...
JEE Main 2019 (Online) 9th April Morning Slot
If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is :...
JEE Main 2019 (Online) 8th April Evening Slot
The tangent and the normal lines at the point ( $$\sqrt 3 $$, 1) to the circle x2 + y2 = 4 and the x-axis form a triangle. The area of this triangle ...
JEE Main 2019 (Online) 8th April Morning Slot
The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 16, by the lines, x + y = n, n $$ \in $$ N, where N is the se...
JEE Main 2019 (Online) 12th January Evening Slot
If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O...
JEE Main 2019 (Online) 12th January Morning Slot
If a variable line, 3x + 4y – $$\lambda $$ = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x – 2y + 78 = 0 are on its opp...
JEE Main 2019 (Online) 12th January Morning Slot
Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2 – 6x – 6y + 14 = 0 respectively. If P and Q are the points of inter...
JEE Main 2019 (Online) 11th January Morning Slot
The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distan...
JEE Main 2019 (Online) 11th January Morning Slot
A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of thi...
JEE Main 2019 (Online) 11th January Morning Slot
Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through t...
JEE Main 2019 (Online) 10th January Evening Slot
If the area of an equilateral triangle inscribed in the circle x2 + y2 + 10x + 12y + c = 0 is $$27\sqrt 3 $$ sq units then c is equal to :...
JEE Main 2019 (Online) 10th January Morning Slot
If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is :...
JEE Main 2019 (Online) 9th January Evening Slot
If the circles x2 + y2 $$-$$ 16x $$-$$ 20y + 164 = r2   and  (x $$-$$ 4)2 + (y $$-$$ 7)2 = 36 intersect at two distinct points, ...
JEE Main 2019 (Online) 9th January Morning Slot
Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :
JEE Main 2018 (Online) 16th April Morning Slot
If a circle C, whose radius is 3, touches externally the circle, $${x^2} + {y^2} + 2x - 4y - 4 = 0$$ at the point (2, 2), then the length of the inte...
JEE Main 2018 (Offline)
If the tangent at (1, 7) to the curve x2 = y - 6 touches the circle x2 + y2 + 16x + 12y + c = 0, then the value of c is :...
JEE Main 2018 (Online) 15th April Evening Slot
The tangent to the circle C1 : x2 + y2 $$-$$ 2x $$-$$ 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C2 whose center is (3, $$-$...
JEE Main 2018 (Online) 15th April Morning Slot
A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $$y - 4x + 3 = 0,$$ then its radius is equal to :
JEE Main 2017 (Online) 9th April Morning Slot
A line drawn through the point P(4, 7) cuts the circle x2 + y2 = 9 at the points A and B. Then PA⋅PB is equal to :
JEE Main 2017 (Online) 9th April Morning Slot
The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60o. If the area of the quadrilateral is $$4\sqrt 3 $$, t...
JEE Main 2017 (Online) 8th April Morning Slot
If a point P has co-ordinates (0, $$-$$2) and Q is any point on the circle, x2 + y2 $$-$$ 5x $$-$$ y + 5 = 0, then the maximum value of (PQ)2 is : ...
JEE Main 2017 (Online) 8th April Morning Slot
If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $${\cos ^{ - 1}}\left( {{1 \ove...
JEE Main 2017 (Offline)
The radius of a circle, having minimum area, which touches the curve y = 4 – x2 and the lines, y = |x| is :
JEE Main 2016 (Online) 10th April Morning Slot
Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight lines x − y = 1 and 2x + y = 3 ...
JEE Main 2016 (Online) 9th April Morning Slot
A circle passes through (−2, 4) and touches the y-axis at (0, 2). Which one of the following equations can represent a diameter of this circle?
JEE Main 2016 (Offline)
The centres of those circles which touch the circle, $${x^2} + {y^2} - 8x - 8y - 4 = 0$$, externally and also touch the $$x$$-axis, lie on :
JEE Main 2016 (Offline)
If one of the diameters of the circle, given by the equation, $${x^2} + {y^2} - 4x + 6y - 12 = 0,$$ is a chord of a circle $$S$$, whose centre is at $...
JEE Main 2015 (Offline)
Locus of the image of the point $$(2, 3)$$ in the line $$\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,\,k \in R,$$ is a :
JEE Main 2015 (Offline)
The number of common tangents to the circles $${x^2} + {y^2} - 4x - 6x - 12 = 0$$ and $${x^2} + {y^2} + 6x + 18y + 26 = 0,$$ is :
JEE Main 2014 (Offline)
Let $$C$$ be the circle with centre at $$(1, 1)$$ and radius $$=$$ $$1$$. If $$T$$ is the circle centred at $$(0, y)$$, passing through origin and tou...
JEE Main 2013 (Offline)
The circle passing through $$(1, -2)$$ and touching the axis of $$x$$ at $$(3, 0)$$ also passes through the point :
AIEEE 2012
The length of the diameter of the circle which touches the $$x$$-axis at the point $$(1, 0)$$ and passes through the point $$(2, 3)$$ is :
AIEEE 2011
The two circles x2 + y2 = ax, and x2 + y2 = c2 (c > 0) touch each other if :
AIEEE 2010
The circle $${x^2} + {y^2} = 4x + 8y + 5$$ intersects the line $$3x - 4y = m$$ at two distinct points if :
AIEEE 2009
Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the poi...
AIEEE 2009
If $$P$$ and $$Q$$ are the points of intersection of the circles $${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0$$ and $${x^2} + {y^2} + 2x + 2y - {p^2} = 0$$...
AIEEE 2008
The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is :
AIEEE 2008
The differential equation of the family of circles with fixed radius $$5$$ units and centre on the line $$y = 2$$ is :
AIEEE 2007
Consider a family of circles which are passing through the point $$(-1, 1)$$ and are tangent to $$x$$-axis. If $$(h, k)$$ are the coordinate of the ce...
AIEEE 2006
If the lines $$3x - 4y - 7 = 0$$ and $$2x - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi $$ square units, the equation of the circle is...
AIEEE 2006
Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of the chords of the circle $$C$$ t...
AIEEE 2005
A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is :
AIEEE 2005
If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = {p^2}$$ orthogonally, then the equation of the locus of its cent...
AIEEE 2005
If the circles $${x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0$$ and $${x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, = 0$$ intersect in two ditinct ...
AIEEE 2005
If the pair of lines $$a{x^2} + 2\left( {a + b} \right)xy + b{y^2} = 0$$ lie along diameters of a circle and divide the circle into four sectors such ...
AIEEE 2004
A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is :
AIEEE 2004
If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $$10\,\pi $$, then the equation of the circle is :
AIEEE 2004
If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = 4$$ orthogonally, then the locus of its centre is :
AIEEE 2004
Intercept on the line y = x by the circle $${x^2}\, + \,{y^2} - 2x = 0$$ is AB. Equation of the circle on AB as a diameter is :
AIEEE 2003
If the two circles $${(x - 1)^2}\, + \,{(y - 3)^2} = \,{r^2}$$ and $$\,{x^2}\, + \,{y^2} - \,8x\, + \,2y\, + \,\,8\,\, = 0$$ intersect in two distinct...
AIEEE 2003
The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :
AIEEE 2002
If the chord y = mx + 1 of the circle $${x^2}\, + \,{y^2} = 1$$ subtends an angle of measure $${45^ \circ }$$ at the major segment of the circle then ...
AIEEE 2002
The centre of the circle passing through (0, 0) and (1, 0) and touching the circle $${x^2}\, + \,{y^2} = 9$$ is :
AIEEE 2002
The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $$3$$$$a$$ is :
AIEEE 2002
The centres of a set of circles, each of radius 3, lie on the circle $${x^2}\, + \,{y^2} = 25$$. The locus of any point in the set is :

Numerical

JEE Main 2024 (Online) 9th April Morning Shift
Let the centre of a circle, passing through the points $$(0,0),(1,0)$$ and touching the circle $$x^2+y^2=9$$, be $$(h, k)$$. Then for all possible val...
JEE Main 2024 (Online) 30th January Evening Shift
Consider two circles $$C_1: x^2+y^2=25$$ and $$C_2:(x-\alpha)^2+y^2=16$$, where $$\alpha \in(5,9)$$. Let the angle between the two radii (one to each ...
JEE Main 2024 (Online) 29th January Morning Shift
Equations of two diameters of a circle are $$2 x-3 y=5$$ and $$3 x-4 y=7$$. The line joining the points $$\left(-\frac{22}{7},-4\right)$$ and $$\left(...
JEE Main 2024 (Online) 27th January Evening Shift
Consider a circle $$(x-\alpha)^2+(y-\beta)^2=50$$, where $$\alpha, \beta>0$$. If the circle touches the line $$y+x=0$$ at the point $$P$$, whose dista...
JEE Main 2023 (Online) 12th April Morning Shift
Two circles in the first quadrant of radii $$r_{1}$$ and $$r_{2}$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the l...
JEE Main 2023 (Online) 8th April Morning Shift
Consider a circle $$C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5$$. Let its mirror image in the line $$y=2 x+1$$ be another circle $$C_{2}: 5 x^{2}+5 y^{2}-10 ...
JEE Main 2023 (Online) 6th April Morning Shift
Let the point $$(p, p+1)$$ lie inside the region $$E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$$. If the set of all value...
JEE Main 2023 (Online) 6th April Morning Shift
A circle passing through the point $$P(\alpha, \beta)$$ in the first quadrant touches the two coordinate axes at the points $$A$$ and $$B$$. The poin...
JEE Main 2023 (Online) 30th January Evening Shift
Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $\mathrm{O}$ be ...
JEE Main 2023 (Online) 29th January Evening Shift
A circle with centre (2, 3) and radius 4 intersects the line $$x+y=3$$ at the points P and Q. If the tangents at P and Q intersect at the point $$S(\a...
JEE Main 2023 (Online) 25th January Evening Shift
Points P($$-$$3, 2), Q(9, 10) and R($$\alpha,4$$) lie on a circle C and PR as its diameter. The tangents to C at the points Q and R intersect at the p...
JEE Main 2022 (Online) 29th July Evening Shift
Let $$A B$$ be a chord of length 12 of the circle $$(x-2)^{2}+(y+1)^{2}=\frac{169}{4}$$. If tangents drawn to the circle at points $$A$$ and $$B$$ int...
JEE Main 2022 (Online) 29th July Evening Shift
$$\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$$ and $$T=\left\{(x, y) \in \mathbb{R} \time...
JEE Main 2022 (Online) 29th July Morning Shift
Let the mirror image of a circle $$c_{1}: x^{2}+y^{2}-2 x-6 y+\alpha=0$$ in line $$y=x+1$$ be $$c_{2}: 5 x^{2}+5 y^{2}+10 g x+10 f y+38=0$$. If $$\mat...
JEE Main 2022 (Online) 25th July Evening Shift
If the circles $${x^2} + {y^2} + 6x + 8y + 16 = 0$$ and $${x^2} + {y^2} + 2\left( {3 - \sqrt 3 } \right)x + 2\left( {4 - \sqrt 6 } \right)y = k + 6\sq...
JEE Main 2022 (Online) 28th June Evening Shift
If one of the diameters of the circle $${x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0$$ is a chord of the circle $${(x - 2\sqrt 2 )^2} + {(y - 2\sq...
JEE Main 2022 (Online) 28th June Morning Shift
Let the lines $$y + 2x = \sqrt {11} + 7\sqrt 7 $$ and $$2y + x = 2\sqrt {11} + 6\sqrt 7 $$ be normal to a circle $$C:{(x - h)^2} + {(y - k)^2} = {r^...
JEE Main 2022 (Online) 27th June Evening Shift
Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L2 ...
JEE Main 2022 (Online) 27th June Morning Shift
A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x $$-...
JEE Main 2022 (Online) 25th June Morning Shift
Let the abscissae of the two points P and Q be the roots of $$2{x^2} - rx + p = 0$$ and the ordinates of P and Q be the roots of $${x^2} - sx - q = 0$...
JEE Main 2022 (Online) 24th June Evening Shift
Let a circle C : (x $$-$$ h)2 + (y $$-$$ k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such th...
JEE Main 2021 (Online) 31st August Evening Shift
Let B be the centre of the circle x2 + y2 $$-$$ 2x + 4y + 1 = 0. Let the tangents at two points P and Q on the circle intersect at the point A(3, 1). ...
JEE Main 2021 (Online) 31st August Morning Shift
If the variable line 3x + 4y = $$\alpha$$ lies between the two circles (x $$-$$ 1)2 + (y $$-$$ 1)2 = 1 and (x $$-$$ 9)2 + (y $$-$$ 1)2 = 4, without in...
JEE Main 2021 (Online) 27th August Evening Shift
Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C1($$\alpha$$, $...
JEE Main 2021 (Online) 27th August Morning Shift
Let the equation x2 + y2 + px + (1 $$-$$ p)y + 5 = 0 represent circles of varying radius r $$\in$$ (0, 5]. Then the number of elements in the set S = ...
JEE Main 2021 (Online) 26th August Morning Shift
The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circl...
JEE Main 2021 (Online) 17th March Morning Shift
The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other circle for the given circles'...
JEE Main 2021 (Online) 24th February Evening Shift
Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point ($$-$$5, 0). If the locus of the point P is a...
JEE Main 2021 (Online) 24th February Evening Shift
If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x $$-$$ 2)2 + (y $$-$$ 3)2 = 25 at the point (5, ...
JEE Main 2021 (Online) 24th February Morning Shift
If one of the diameters of the circle x2 + y2 - 2x - 6y + 6 = 0 is a chord of another circle 'C', whose center is at (2, 1), then its radius is ______...
JEE Main 2020 (Online) 4th September Evening Slot
Let PQ be a diameter of the circle x2 + y2 = 9. If $$\alpha $$ and $$\beta $$ are the lengths of the perpendiculars from P and Q on the straight line,...
JEE Main 2020 (Online) 3rd September Morning Slot
The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is _______...
JEE Main 2020 (Online) 2nd September Morning Slot
The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x2 + y2 – 2x – 4y + 4 = 0 at two distinct points is ______...
JEE Main 2020 (Online) 9th January Evening Slot
If the curves, x2 – 6x + y2 + 8 = 0 and x2 – 8y + y2 + 16 – k = 0, (k > 0) touch each other at a point, then the largest value of k is ______....
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