Q JEE Main 2024

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Consider two circles $\mathrm{C}_1: \mathrm{x}^2+\mathrm{y}^2=25$ and $\mathrm{C}_2:(\mathrm{x}-\alpha)^2+y^2=16$, where $\alpha \in(5,9)$. Let the angle between the two radii (one to each...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let the locus of the mid points of the chords of circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line...

JEE Main Mathematics Easy

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Q JEE-Main 2024

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...

JEE Main Mathematics Easy

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Q JEE MAIN 2026

Let $y=x$ be the equation of a chord of the circle $\mathrm{C}_1$ (in the closed half-plane $x \geq 0$ )...

JEE Main Physics Easy

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Q JEE MAIN 2026

Let the circle $x^2+y^2=4$ intersect $x$-axis at the points $\mathrm{A}(\mathrm{a}, 0), \mathrm{a}>0$ and $\mathrm{B}(\mathrm{b}, 0)$. Let $...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let a circle of radius 4 pass through the origin O , the points $\mathrm{A}(-\sqrt{3} a, 0)$ and $\mathrm{B}(0,-\sqrt{2} b)$,...

JEE Main Mathematics Easy

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Q JEE MAIN_2026_

Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4 x-2 y-4=0$ intersect at two...

JEE Main Chemistry Medium

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Q JEE MAIN 2026

If P is a point on the circle $x^2+y^2=4, \mathrm{Q}$ is a point on the straight line $5 x+y+2=0$ and...

JEE Main Mathematics Hard

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Q JEE MAIN 2026

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4 x-6 y-3=0$ at the points A and B...

JEE Main Mathematics Medium

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Q JEE MAIN_2019_

If the circles $x^2+y^2+5 K x+2 y+K=0$ and $2\left(x^2+\right. \left.y^2\right)+2 K x+3 y-1=0,(K \in R)$, intersect at the points $P$...

JEE Main Mathematics Medium

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Q JEE MAIN_2019_

The line x = y touches a circle at the point (1, 1). If the circle also passes through the...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points P and Q, then the locus...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through...

JEE Main Mathematics Medium

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Q JEE Main 2020

The number of integral values of $k$ for which the line, $3 x+4 y=k$ intersects the circle, $...

JEE Main Mathematics Medium

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Q JEE-MAIN 2019

Let $C 1$ and $C 2$ be the centres of the circles $x^2+y^2-2 x-2 y-2=0$ and $x^2+y^2-6 x-6 y+14=0$ respectively....

JEE Main Mathematics Easy

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Q JEE-MAIN 2019

If a variable line, $3 x+4 y-\lambda=0$ is such that the two circles $x^2+y^2-2 x-2 y+1=0$ and $x^2+y^2-18 x-2 y+78=0$...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

If the circles $x^2+y^2-16 x-20 y+164=r^2$ and $(x-4)^2+(y-7)^2=36$ intersect at two distinct points, then

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If a circle of radius R pases through the origin O and intersects the coordinate axes at A and B,...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If the circles $x^2+y^2-16 x-20 y+164=r^2$ and $(x-4)^2+(y-7)^2=36$ intersect at two distinct points, then

JEE Main Mathematics Easy

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Q JEE MAIN 2019

A square is inscribed in the circle $x^2+y^2-6 x+8 y-103=0$ with its sides parallel to the coordinate axes. Then the...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

The straight line $x+2 y=1$ meets the coordinate axes at A and B. A circle is drawn through A, B...

JEE Main Chemistry Medium

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Q JEE MAIN 2020

A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the...

JEE Main Mathematics Easy

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Q JEE-MAIN 2020

If the length of the chord of the circle, $x^2+y^2=r^2(r>0)$ along the line, $y-2 x=3$ is $r$, then $r^2$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y$-axis and lie...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

Let $P Q$ be a diameter of the circle $x^2+y^2=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

The circle passing through the intersection of the circles, $x^2+y^2-6 x=0$ and $x^2+y^2-4 y=0$, having its centre on the line,...

JEE Main Mathematics Hard

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Q JEE Main 2019

If the line $3 x+4 y-24=0$ intersects the $x$-axis at the point $A$ and the $y$-axis at the point $B$,...

JEE Main Mathematics Easy

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Q JEE Main 2019

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4 x-6 y=12$ externally at the point $(1$,...

JEE Main Mathematics Easy

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Q JEE MAIN 2020

If a line, y=m x+c is a tangent to the circle $\ldots(x-3)^2+y^2=1$ and it is perpendicular to a line $L_1$,...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

The common tangent to the circles $x^2+y^2=4$ and $x^2+y^2+6 x+8 y-24=0$ also passes through the point :

JEE Main Mathematics Easy

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Q JEE Main 2021

Let $B$ be the centre of the circle $x^2+y^2-2 x+4 y+1=0$. Let the tangents at two point $P$ and $Q$...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the variable line $3 x+4 y=\alpha$ lies between the two circles $(x-1)^2+(y-1)^2=1$ and $(x-9)^2+(y-1)^2=4$, without intercepting a chord on...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The line $12 x \cos \theta+5 y \sin \theta=60$ is tangent to which of the following curves?

JEE Main Mathematics Easy

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Q JEE MAIN 2019

\text { If the area of an equilateral triangle inscribed in the circle, } x^2+y^2+10 x+12 y+c=0 \text { is...

JEE Main Mathematics Medium

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Q JEE Main 2021

The locus of a point, which moves such that the sum of squares of its distances from the points $(0,0),(1,0)$,...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let the equation $x^2+y^2+p x+(1-p) y+5=0$ represent circles of varying radius $r \in(0,5]$. Then the number of elements in the...

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

Two circles each of radius 5 units touch each other at the point $(1,2)$. If the equation of their common...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

A circle $C$ touches the line $x=2 y$ at the point $(2,1)$ and intersects the circle $...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16$, by the lines, $...

JEE Main Mathematics Medium

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Q JEE Main 2019

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis...

JEE Main Mathematics Hard

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Q JEE MAIN 2021 S2

Consider a circle $C$ which touches the $y$-axis at $(0,6)$ and cuts off an intercept $6 \sqrt{5}$ on the $x$-axis....

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

A spherical gas balloon of radius 16 meter subtends an angle $60^{\circ}$ at the eye of the observer A while...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^2+y^2=4$ and the $x$-axis form a...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

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 $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2 x-6 y+6=0$. If these tangents touch the circle...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle,...

JEE Main Mathematics Easy

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Q JEE MAIN_2021

Let the circle $\mathrm{S} ; 36 x^2+36 y^2-108 x+120 y+C=0$ be such that it neither intersects nor touches the coordinate...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

Let the tangent to the circle $x^2+y^2=25$ at the point $R(3,4)$ meet $x$-axis and $y$-axis at point $P$ and $Q$,...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $S_1: x^2+y^2=9$ and $S_2:(x-2)^2+y^2=1$. Then the locus of center of a variable circle $S$ which touches $S_1$ internally and...

JEE Main Mathematics Medium

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Q _JEE MAIN_2021_

The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$...

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

Choose the incorrect statement about the two circles whose equations are given below. $$ \begin{aligned} & x^2+y^2-10 x-10 y+41=0 \text...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If one of the diameters of the circle $x^2+y^2-2 x-6 y+6=0$ is a chord of another circle ' $C$ ',...

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

The line 2x – y + 1 = 0 is a tangent to the circle at the point (2, 5)...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x^2+y^2+a x+2 a y+c=0,(a< 0)$ be $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $r_1$ and $r_2$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the area of the triangle formed by the positive xaxis, the normal and the tangent to the circle (...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let a point P be such that its distance from the point $(5,0)$ is thrice the distance of P from...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

For the four circles $\mathrm{M}, \mathrm{N}, \mathrm{O}$ and P , following four equations are given : Circle $\mathrm{M}: \mathrm{x}^2+\mathrm{y}^2=1$ Circle...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Choose the correct statement about two circles whose equation are given below : $...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

Let A B C D be a square of side of unit length. Let a circle $C_1$ centered at $A$...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let R = {(P,Q) | P and Q are at the same distance from the origin} be a relation, then...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

In the circle given below, let $\mathrm{OA}=1$ unit, $\mathrm{OB}=13$ unit and $\mathrm{PQ} \perp \mathrm{OB}$. Then, the area of the triangle...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let a circle $C$ of radius 5 lie below the $x$-axis. The line $L_1=4 x+3 y-2$ passes through the centre...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

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 Mathematics Easy

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Q JEE MAIN 2022

Let a circle C : $(x-h)^2+(y-k)^2=r^2, k>0$,, touch the x-axis at (1, 0). If the line x + y =...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The set of values of $k$ for which the circle $C: 4 x^2+4 y^2-12 x+8 y+k=0$ lies inside the fourth...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let a triangle $A B C$ be inscribed in the circle $x^2-\sqrt{2}(x+y)+y^2=0$ such that $\angle B A C=\frac{\pi}{2}$. If the...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let the tangent to the circle $C_1: x^2+y^2=2$ at the point $M(-1,1)$ intersect the circle $C_2:(x-3)^2+(y-2)^2=5$ at two distinct points...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let the lines $y+2 x=\sqrt{11}+7 \sqrt{7}$ and $2 y+x=2 \sqrt{11}+6 \sqrt{7}$ be normal to a circle $C:(x-h)^2+(y-k)^2=r^2$. If the line...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

A rectangle R with end points of the one of its dies as (1, 2) and (3, 6) is inscribed...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

If the tangents drawn at the point $\mathrm{O}(0,0)$ and $\mathrm{P}(1+\sqrt{5}, 2)$ on the circle $x^2+y^2-2 x-4 y=0$ intersect at the...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\frac{d y}{d x}=\frac{a x-b y+a}{b x+c y+a}$, where $a, b, c$ are constants, represent a circle passing through the...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $A B$ is not a...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let a circle $C$ touch the lines $L_1: 4 x-3 y+K_1=0$ and $L_2: 4 x-3 y+K_2=0, K_1, K_2 \in R$....

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let $O$ be the origin and $O P$ and $O Q$ be the tangents to the circle $x^2+y^2-6 x+4 y+8=0$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let the centre of a circle C be (α, β) and its radius r < 8. Let 3x + 4y...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

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...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k, 3 k)$ lie on a circle of radius $r$, then $10 k+r^2$...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

$$ \text { The number of common tangents, to the circles } x^2+y^2-18 x-15 y+131=0 \text { and } x^2+y^2-6...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off...

JEE Main Chemistry Medium

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Q JEE MAIN 2023

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 $...

JEE Main Mathematics Easy

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Q JEE-Main 2023

A circle passing through the point P($\alpha, \beta$) in the first quadrant touches the two coordinate axes at the points...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

A line segment AB of length l moves such that the points A and B remain on the periphery of...

JEE Main Mathematics Easy

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Q JEE MAIN

Let the centre of a circle, passing through the point (0,0),(1,0) and touching the circle $x^2+y^2$ =9, be (h,k). Then...

JEE Main Mathematics Medium

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Q JEE-Main

A circle in inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

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 $y= x+3$....

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the...

JEE Main Mathematics Medium

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Q JEE-Main 2024

Let the line $\mathrm{L}: \sqrt{2} \mathrm{x}+\mathrm{y}=\alpha$ pass through the point of the intersection P (in the first quadrant) of the...

JEE Main Mathematics Medium

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Q JEE-Main 2024

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...

JEE Main Mathematics Hard

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Q JEE MAIN

Equation of two diameters of a circle are 2x-3y=5 and 3x-4y=7. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7}, 3\right)$...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If one of the diameters of the circle $x^2+y^2-10 x+4 y+13=0$ is a chord of another circle $C$, whose center...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Four distinct points (2k,3k),(1,0),(0,1) and (0,0) lie on a circle for k equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let a circle $C$ pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3 x+2 y+2=0$. Then...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let ${C_1}$ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let ${C_2}$ be...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the line $x + y = 1$ meet the circle ${x^2} + {y^2} = 4$ at the points A...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the equation of the circle, which touches x-axis at the point $(a,0),a > 0$ and cuts off an intercept...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let circle C be the image of ${x^2} + {y^2} - 2x + 4y - 4 - 0$ in the...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be...

JEE Main Mathematics Medium

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