Question

Let

 $A=\left\{\left(x,y\right)\in R\times R\mid 2x^2+2y^2-2x-2y=1\right\}$

$B=\left\{\left(x,y\right)\in R\times R\mid 4x^2+4y^2-16y+7=0\right\}$ and

$C=\left\{\left(x,y\right)\in R\times R\mid x^2+y^2-4x-2y+5\le r^2\right\}$. Then the minimum value of $\left|r\right|$ such that $A\cup B\subseteq C$ is equal to

JEE Main 2021 (27 Jul Shift 1)

Options

• A: $\frac{3+\sqrt{10}}{2}$
• B: $\frac{2+\sqrt{10}}{2}$
• C: $\frac{3+2\sqrt{5}}{2}$
• D: $1+\sqrt{5}$

Question

A point $P$ moves so that the sum of squares of its distances from the points $\left(1,2\right)$ and $\left(-2,1\right)$ is $14$. Let $f\left(x,y\right)=0$ be the locus of $P$, which intersects the $x$-axis at the points $A,B$ and the $y$-axis at the point $C,D$. Then the area of the quadrilateral $ACBD$ is equal to

JEE Main 2022 (26 Jul Shift 1)

Options

• A: $\frac{9}{2}$
• B: $\frac{3\sqrt{17}}{2}$
• C: $\frac{3\sqrt{17}}{4}$
• D: $9$

Question

Let $C$ be a circle passing through the points $A\left(2,-1\right)$ and $B\left(3,4\right)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle ${\left(x-5\right)}^2+{\left(y-1\right)}^2=\frac{13}{2}$, then $r^2$ is equal to

JEE Main 2022 (26 Jun Shift 1)

Options

• A: $32$
• B: $\frac{65}{2}$
• C: $\frac{61}{2}$
• D: $30$

Question

If the circles $x^2+y^2+6x+8y+16=0$ and $x^2+y^2+2\left(3-\sqrt{3}\right)x+2\left(4-\sqrt{6}\right)y=k+6\sqrt{3}+8\sqrt{6}$, $k>0$, touch internally at the point $P\left(\alpha ,\beta \right)$, then ${\left(\alpha +\sqrt{3}\right)}^2+{\left(\beta +\sqrt{6}\right)}^2$ is equal to _______.

JEE Main 2022 (25 Jul Shift 2)

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Question

A rectangle $R$ with end points of the one of its sides as $\left(1,2\right)$ and $\left(3,6\right)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of $R$ is _____.

JEE Main 2022 (27 Jun Shift 1)

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Question

Let the mirror image of a circle $c_1:x^2+y^2-2x-6y+\alpha =0$  in line $y=x+1$ be $c_2:5x^2+5y^2+10gx$ $+10fy+38=0$. If $r$ is the radius of circle $c_2$, then $\alpha +6r^2$ is equal to ______

JEE Main 2022 (29 Jul Shift 1)

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Question

Let the abscissae of the two points $P$ and $Q$ be the roots of $2x^2-rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of $x^2-sx-q=0$. If the equation of the circle described on $PQ$ as diameter is $2\left(x^2+y^2\right)-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.

JEE Main 2022 (25 Jun Shift 1)

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Question

The locus of the middle points of the chords of the circle $C_1: {\left(x-4\right)}^2+{\left(y-5\right)}^2=4$ which subtend an angle ${\theta }_i$ at the centre of the circle $C_i$, is a circle of radius $r_i$. If ${\theta }_1=\frac{\mathrm{\pi }}{3}$, ${\theta }_3=\frac{2\mathrm{\pi }}{3}$ and ${r_1}^2={r_2}^2+{r_3}^2$, then ${\theta }_2$ is equal to

JEE Main 2023 (24 Jan Shift 2)

Options

• A: $\frac{\mathrm{\pi }}{4}$
• B: $\frac{3\mathrm{\pi }}{4}$
• C: $\frac{\mathrm{\pi }}{6}$
• D: $\frac{\mathrm{\pi }}{2}$

Question

The points of intersection of the line $ax + by=0$, $(\mathrm{a}\ne \mathrm{b})$ and the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}=0$ are $A(\alpha ,0)$ and $B(1,\beta )$. The image of the circle with $AB$ as a diameter in the line $\mathrm{x}+\mathrm{y}+2=0$ is :

JEE Main 2023 (25 Jan Shift 1)

Options

• A: $x^2+y^2+5x+5y+12=0$
• B: $x^2+y^2+3x+5y+8=0$
• C: $x^2+y^2+3x+3y+4=0$
• D: $x^2+y^2-5x-5y+12=0$

Question

The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2},\frac{1-a}{2}\right)$ on the circle $2x^2+2y^2-(1+a)x-(1-a)y=0$, is equal to :

JEE Main 2023 (31 Jan Shift 2)

Options

• A: $(8,\infty )$
• B: $(0,4]$
• C: $(4,\infty )$
• D: $(2,12]$

Question

Consider ellipses $E_k:k{x}^2+k^2{y}^2=1,k=1,2,\dots ,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$. If $r_k$ is the radius of the circle $C_k$, then the value of $\sum _{k=1}^{20}\frac{1}{r_k^2}$ is

JEE Main 2023 (11 Apr Shift 1)

Options

• A: $3080$
• B: $2870$
• C: $3210$
• D: $3320$

Question

Consider a circle $C_1 : x{ }^2+y^2 – 4x – 2y = \alpha – 5.$ Let its mirror image in the line $y=2x+1$ be another circle $C_2:5x^2+5y^2 –10fx – 10gy+ 36 = 0.$  Let $r$ be the radius of $C_2$ . Then $\alpha + r$ is equal to $\_\_\_\_\_\_\_\_$

JEE Main 2023 (08 Apr Shift 1)

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Question

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 vertices of the square, then $\mathbf{\Sigma}\left(x_i^2+y_i^2\right)$ is equal to:

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 148
• B: 152
• C: 160
• D: 156

Question

Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $\mathrm{C}$ from the point $(5,5)$ is :

JEE Main 2024 (05 Apr Shift 1)

Options

• A: $2 \sqrt{2}$
• B: $4 \sqrt{2}$
• C: 4
• D: 5

Question

Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment $\mathrm{AB}$ and the point $\mathrm{F}$ is on the diagonal $\mathrm{AC}$. Then the radius $\mathrm{r}$ of the circle passing through the point $\mathrm{F}$ and touching the line segments $\mathrm{BC}$ and $\mathrm{CD}$ satisfies:

JEE Main 2024 (05 Apr Shift 2)

Options

• A: $r=0$
• B: $2 r^2-4 r+1=0$
• C: $2 r^2-8 r+7=0$
• D: $r^2-8 r+8=0$

Question

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 line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2: 1$, then $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

JEE Main 2024 (08 Apr Shift 1)

Options

• A: 125
• B: 130
• C: 110
• D: 145

Question

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 values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to_________

JEE Main 2024 (09 Apr Shift 1)

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Question

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 circle) drawn from one of the intersection points of ${\mathrm{C}}_1$and ${\mathrm{C}}_2$ be ${\sin }^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha \beta {)}^2$ equals _________.

JEE Main 2024 (30 Jan Shift 2)

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