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$$\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$$ and $$T=\lef...
JEE Main 2022 (Online) 29th July Evening Shift
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Let $$A B$$ be a chord of length 12 of the circle $$(x-2)^{2}+(y+1)^{2}=\frac{169}{4}$$. If tangents drawn to the circle...
JEE Main 2022 (Online) 29th July Evening Shift
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Let the mirror image of a circle $$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}+1...
JEE Main 2022 (Online) 29th July Morning Shift
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If the circles $${x^2} + {y^2} + 6x + 8y + 16 = 0$$ and $${x^2} + {y^2} + 2\left( {3 - \sqrt 3 } \right)x + 2\left( {4 -...
JEE Main 2022 (Online) 25th July Evening Shift
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Let a circle C : (x $$-$$ h)2 + (y $$-$$ k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects t...
JEE Main 2022 (Online) 24th June Evening Shift
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Let the abscissae of the two points P and Q be the roots of $$2{x^2} - rx + p = 0$$ and the ordinates of P and Q be the ...
JEE Main 2022 (Online) 25th June Morning Shift
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A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a di...
JEE Main 2022 (Online) 27th June Morning Shift
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Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle...
JEE Main 2022 (Online) 27th June Evening Shift
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Let the lines $$y + 2x = \sqrt {11} + 7\sqrt 7 $$ and $$2y + x = 2\sqrt {11} + 6\sqrt 7 $$ be normal to a circle $$C:{...
JEE Main 2022 (Online) 28th June Morning Shift
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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 $${(...
JEE Main 2022 (Online) 28th June Evening Shift
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Let B be the centre of the circle x2 + y2 $$-$$ 2x + 4y + 1 = 0. Let the tangents at two points P and Q on the circle in...
JEE Main 2021 (Online) 31st August Evening Shift
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If the variable line 3x + 4y = $$\alpha$$ lies between the two circles (x $$-$$ 1)2 + (y $$-$$ 1)2 = 1 and (x $$-$$ 9)2 ...
JEE Main 2021 (Online) 31st August Morning Shift
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Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x +...
JEE Main 2021 (Online) 27th August Evening Shift
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Let the equation x2 + y2 + px + (1 $$-$$ p)y + 5 = 0 represent circles of varying radius r $$\in$$ (0, 5]. Then the numb...
JEE Main 2021 (Online) 27th August Morning Shift
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The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), ...
JEE Main 2021 (Online) 26th August Morning Shift
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The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other...
JEE Main 2021 (Online) 17th March Morning Shift
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If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x $$-$$ 2)2 + (y $$...
JEE Main 2021 (Online) 24th February Evening Slot
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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...
JEE Main 2021 (Online) 24th February Evening Slot
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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,...
JEE Main 2021 (Online) 24th February Morning Slot
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Let PQ be a diameter of the circle x2 + y2 = 9. If $$\alpha $$ and $$\beta $$ are the lengths of the perpendiculars from...
JEE Main 2020 (Online) 4th September Evening Slot
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The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touches both the lin...
JEE Main 2020 (Online) 3rd September Morning Slot
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The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x2 + y2 – 2x – 4y + 4 = 0 at...
JEE Main 2020 (Online) 2nd September Morning Slot
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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 larg...
JEE Main 2020 (Online) 9th January Evening Slot
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MCQ (Single Correct Answer)
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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 ...
JEE Main 2022 (Online) 28th July Evening Shift
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The foot of the perpendicular from a point on the circle $$x^{2}+y^{2}=1, z=0$$ to the plane $$2 x+3 y+z=6$$ lies on whi...
JEE Main 2022 (Online) 28th July Morning Shift
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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 pass...
JEE Main 2022 (Online) 28th July Morning Shift
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For $$\mathrm{t} \in(0,2 \pi)$$, if $$\mathrm{ABC}$$ is an equilateral triangle with vertices $$\mathrm{A}(\sin t,-\cos ...
JEE Main 2022 (Online) 28th July Morning Shift
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A circle $$C_{1}$$ passes through the origin $$\mathrm{O}$$ and has diameter 4 on the positive $$x$$-axis. The line $$y=...
JEE Main 2022 (Online) 27th July Evening Shift
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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 li...
JEE Main 2022 (Online) 27th July Morning Shift
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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 $...
JEE Main 2022 (Online) 26th July Evening Shift
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Consider three circles: $${C_1}:{x^2} + {y^2} = {r^2}$$ $${C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}$$ $${C_3}:{(x - 2)^2} ...
JEE Main 2022 (Online) 30th June Morning Shift
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Let a, b and c be the length of sides of a triangle ABC such that $${{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \ov...
JEE Main 2022 (Online) 25th June Morning Shift
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Let a circle C touch the lines $${L_1}:4x - 3y + {K_1} = 0$$ and $${L_2} = 4x - 3y + {K_2} = 0$$, $${K_1},{K_2} \in R$$....
JEE Main 2022 (Online) 25th June Morning Shift
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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 Evening Shift
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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 i...
JEE Main 2022 (Online) 26th June Morning Shift
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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 an...
JEE Main 2022 (Online) 27th June Evening Shift
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If the tangents drawn at the points $$O(0,0)$$ and $$P\left( {1 + \sqrt 5 ,2} \right)$$ on the circle $${x^2} + {y^2} - ...
JEE Main 2022 (Online) 28th June Morning Shift
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Let a triangle ABC be inscribed in the circle $${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$ such that $$\angle BAC = {\pi \ov...
JEE Main 2022 (Online) 29th June Evening Shift
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Let the tangent to the circle C1 : x2 + y2 = 2 at the point M($$-$$1, 1) intersect the circle C2 : (x $$-$$ 3)2 + (y $$-...
JEE Main 2022 (Online) 29th June Morning Shift
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Let Z be the set of all integers,$$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $$$$B = \{ (x,y) \in Z \times...
JEE Main 2021 (Online) 27th August Evening Shift
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A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C1 : x2 + y2 + 2y $$-$$ 5 = 0 at two po...
JEE Main 2021 (Online) 26th August Evening Shift
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If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point ($$-$$30, 0) and is tangent t...
JEE Main 2021 (Online) 26th August Morning Shift
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Let $$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} - 16...
JEE Main 2021 (Online) 27th July Morning Shift
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Two tangents are drawn from the point P($$-$$1, 1) to the circle x2 + y2 $$-$$ 2x $$-$$ 6y + 6 = 0. If these tangents to...
JEE Main 2021 (Online) 27th July Morning Shift
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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 r...
JEE Main 2021 (Online) 27th July Evening Shift
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Let the circle S : 36x2 + 36y2 $$-$$ 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate ...
JEE Main 2021 (Online) 22th July Evening Shift
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Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point ($$-$$4, 1) a...
JEE Main 2021 (Online) 20th July Evening Shift
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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 in...
JEE Main 2021 (Online) 18th March Evening Shift
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For the four circles M, N, O and P, following four equations are given :Circle M : x2 + y2 = 1Circle N : x2 + y2 $$-$$ 2...
JEE Main 2021 (Online) 18th March Morning Shift
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Choose the correct statement about two circles whose equations are given below :x2 + y2 $$-$$ 10x $$-$$ 10y + 41 = 0x2 +...
JEE Main 2021 (Online) 18th March Morning Shift
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Two tangents are drawn from a point P to the circle x2 + y2 $$-$$ 2x $$-$$ 4y + 4 = 0, such that the angle between these...
JEE Main 2021 (Online) 17th March Evening Shift
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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. ...
JEE Main 2021 (Online) 17th March Evening Shift
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Choose the incorrect statement about the two circles whose equations are given below :x2 + y2 $$-$$ 10x $$-$$ 10y + 41 =...
JEE Main 2021 (Online) 17th March Morning Shift
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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 $$-$$ ...
JEE Main 2021 (Online) 17th March Morning Shift
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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$${\sqr...
JEE Main 2021 (Online) 16th March Evening Shift
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If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circl...
JEE Main 2021 (Online) 26th February Evening Shift
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Let A(1, 4) and B(1, $$-$$5) be two points. Let P be a point on the circle (x $$-$$ 1)2 + (y $$-$$ 1)2 = 1 such that (PA...
JEE Main 2021 (Online) 26th February Evening Shift
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In the circle given below, let OA = 1 unit, OB = 13 unit and PQ $$ \bot $$ OB. Then, the area of the triangle PQB (in sq...
JEE Main 2021 (Online) 26th February Morning Shift
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If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segm...
JEE Main 2021 (Online) 25th February Evening Shift
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Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be $$...
JEE Main 2021 (Online) 24th February Evening Slot
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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) 5th September Evening Slot
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The circle passing through the intersection of the circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having its centre on ...
JEE Main 2020 (Online) 4th September Evening Slot
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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...
JEE Main 2020 (Online) 9th January Morning Slot
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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 th...
JEE Main 2020 (Online) 8th January Evening Slot
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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 (...
JEE Main 2020 (Online) 7th January Evening Slot
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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 Evening Slot
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A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then the centro...
JEE Main 2019 (Online) 12th April Evening Slot
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If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90o, then the lengt...
JEE Main 2019 (Online) 12th April Morning Slot
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The locus of the centres of the circles, which touch the circle, x2 + y2 = 1 externally, also touch the y-axis and lie...
JEE Main 2019 (Online) 10th April Evening Slot
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The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its radiu...
JEE Main 2019 (Online) 10th April Morning Slot
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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 ...
JEE Main 2019 (Online) 10th April Morning Slot
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A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of th...
JEE Main 2019 (Online) 9th April Evening Slot
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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
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If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the...
JEE Main 2019 (Online) 9th April Morning Slot
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The tangent and the normal lines at the point ( $$\sqrt 3 $$, 1) to the circle x2 + y2 = 4 and the x-axis form a triang...
JEE Main 2019 (Online) 8th April Evening Slot
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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 ...
JEE Main 2019 (Online) 8th April Morning Slot
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If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of th...
JEE Main 2019 (Online) 12th January Evening Slot
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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...
JEE Main 2019 (Online) 12th January Morning Slot
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If a variable line, 3x + 4y – $$\lambda $$ = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18...
JEE Main 2019 (Online) 12th January Morning Slot
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Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one o...
JEE Main 2019 (Online) 11th January Morning Slot
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A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the...
JEE Main 2019 (Online) 11th January Morning Slot
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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 t...
JEE Main 2019 (Online) 11th January Morning Slot
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If the area of an equilateral triangle inscribed in the circle x2 + y2 + 10x + 12y + c = 0 is $$27\sqrt 3 $$ sq units t...
JEE Main 2019 (Online) 10th January Evening Slot
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If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, – 1...
JEE Main 2019 (Online) 10th January Morning Slot
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If the circles x2 + y2 $$-$$ 16x $$-$$ 20y + 164 = r2 and (x $$-$$ 4)2 + (y $$-$$ 7)2 = 36 inte...
JEE Main 2019 (Online) 9th January Evening Slot
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Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, ...
JEE Main 2019 (Online) 9th January Morning Slot
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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...
JEE Main 2018 (Online) 16th April Morning Slot
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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 circ...
JEE Main 2018 (Online) 15th April Evening Slot
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A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $$y - 4x + 3 = 0,$$ then its radiu...
JEE Main 2018 (Online) 15th April Morning Slot
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Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this tr...
JEE Main 2018 (Offline)
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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 (Offline)
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The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60o. If the area of the quadr...
JEE Main 2017 (Online) 9th April Morning Slot
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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
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The equation Im $$\left( {{{iz - 2} \over {z - i}}} \right)$$ + 1 = 0, z $$ \in $$ C, z $$ \ne $$ i represents a part of...
JEE Main 2017 (Online) 9th April Morning Slot
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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 ...
JEE Main 2017 (Online) 8th April Morning Slot
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The radius of a circle, having minimum area, which touches the curve y = 4 – x2 and the lines, y = |x| is :
JEE Main 2017 (Offline)
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Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight l...
JEE Main 2016 (Online) 10th April Morning Slot
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A circle passes through (−2, 4) and touches the y-axis at (0, 2). Which one of the following equations can represent a d...
JEE Main 2016 (Online) 9th April Morning Slot
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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 circ...
JEE Main 2016 (Offline)
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The centres of those circles which touch the circle, $${x^2} + {y^2} - 8x - 8y - 4 = 0$$, externally and also touch the ...
JEE Main 2016 (Offline)
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The number of common tangents to the circles $${x^2} + {y^2} - 4x - 6x - 12 = 0$$ and $${x^2} + {y^2} + 6x + 18y + 26 =...
JEE Main 2015 (Offline)
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Locus of the image of the point $$(2, 3)$$ in the line $$\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0...
JEE Main 2015 (Offline)
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Let $$C$$ be the circle with centre at $$(1, 1)$$ and radius $$=$$ $$1$$. If $$T$$ is the circle centred at $$(0, y)$$, ...
JEE Main 2014 (Offline)
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The circle passing through $$(1, -2)$$ and touching the axis of $$x$$ at $$(3, 0)$$ also passes through the point
JEE Main 2013 (Offline)
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The length of the diameter of the circle which touches the $$x$$-axis at the point $$(1, 0)$$ and passes through the poi...
AIEEE 2012
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The two circles x2 + y2 = ax, and x2 + y2 = c2 (c > 0) touch each other if ...
AIEEE 2011
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The circle $${x^2} + {y^2} = 4x + 8y + 5$$ intersects the line $$3x - 4y - m$$ at two distinct points if
AIEEE 2010
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If $$P$$ and $$Q$$ are the points of intersection of the circles $${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0$$ and $${x^2} +...
AIEEE 2009
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The differential equation of the family of circles with fixed radius $$5$$ units and centre on the line $$y = 2$$ is
AIEEE 2008
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The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is
AIEEE 2008
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Consider a family of circles which are passing through the point $$(-1, 1)$$ and are tangent to $$x$$-axis. If $$(h, k)$...
AIEEE 2007
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Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of th...
AIEEE 2006
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If the lines $$3x - 4y - 7 = 0$$ and $$2x - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi $$ square units,...
AIEEE 2006
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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 ...
AIEEE 2005
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If the circles $${x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0$$ and $${x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, =...
AIEEE 2005
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If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = {p^2}$$ orthogonally, then the equ...
AIEEE 2005
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A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of t...
AIEEE 2005
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Intercept on the line y = x by the circle $${x^2}\, + \,{y^2} - 2x = 0$$ is AB. Equation of the circle on AB as a diamet...
AIEEE 2004
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A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter...
AIEEE 2004
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If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = 4$$ orthogonally, then the locus o...
AIEEE 2004
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If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $$10\,\pi $$, then the e...
AIEEE 2004
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The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the c...
AIEEE 2003
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If the two circles $${(x - 1)^2}\, + \,{(y - 3)^2} = \,{r^2}$$ and $$\,{x^2}\, + \,{y^2} - \,8x\, + \,2y\, + \,\,8\,\, =...
AIEEE 2003
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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 ...
AIEEE 2002
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The equation of a circle with origin as a center and passing thorough equilateral triangle whose median is of length $$3...
AIEEE 2002
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The centre of the circle passing through (0, 0) and (1, 0) and touching the circle $${x^2}\, + \,{y^2} = 9$$ is
AIEEE 2002
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If the chord y = mx + 1 of the circle $${x^2}\, + \,{y^2} = 1$$ subtends an angle of measure $${45^ \circ }$$ at the maj...
AIEEE 2002
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