Circles · JEE Main previous year questions

If $\mathrm{P}(6,1)$ be the orthocentre of the triangle whose vertices are $\mathrm{A}(5,-2)$, $\mathrm{B}(8,3)$ and $\mathrm{C}(h,k)$, then the point $\mathrm{C}$ lies on the circle :

1. A.$x^2+y^2-74=0$
2. B.$x^2+y^2-65=0$
3. C.$x^2+y^2-61=0$
4. D.$x^2+y^2-52=0$

Let the circle $C_1:x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of $C_1$ and $C_2$ intersects the $y$-axis at the point P, then the square of the distance of P from the centre of $C_1$ is:

1. A.4
2. B.6
3. C.2
4. D.1

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :

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

The differential equation of the family of circles passing through the origin and having centre at the line $y=x$ is :

1. A.$(x^2-y^2+2xy)dx=(x^2-y^2+2xy)dy$
2. B.$(x^2+y^2-2xy)dx=(x^2+y^2+2xy)dy$
3. C.$(x^2+y^2+2xy)dx=(x^2+y^2-2xy)dy$
4. D.$(x^2-y^2+2xy)dx=(x^2-y^2-2xy)dy$

Let a line perpendicular to the line $2x-y=10$ touch the parabola $y^2=4(x-9)$ at the point P. The distance of the point P from the centre of the circle $x^2+y^2-14x-8y+56=0$ is _________.

A square is inscribed in the circle $x^2+y^2-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_i,y_i)$ are the vertices of the square, then $\sum (x_i^2+y_i^2)$ is equal to:

1. A.152
2. B.148
3. C.156
4. D.160

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersect the circle $C$ at the points $\mathrm{P}$ and $\mathrm{Q}$. Let $\mathrm{MN}$ be a chord of $C$ of length 2 unit and slope $-1$. Then, a distance (in units) between the chord $\mathrm{PQ}$ and the chord $\mathrm{MN}$ is

1. A.$3-\sqrt{2}$
2. B.$2-\sqrt{3}$
3. C.$\sqrt{2}-1$
4. D.$\sqrt{2}+1$

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 $\mathrm{C}$ and ${\mathrm{C}}^{\prime }$ intersect at two distinct points, is $\mathrm{R}-[\mathrm{a},\mathrm{b}]$, then the point $(8\mathrm{a}+12,16\mathrm{b}-20)$ lies on the curve :

1. A.$x^2+2y^2-5x+6y=3$
2. B.$5x^2-y=-11$
3. C.$x^2-4y^2=7$
4. D.$6x^2+y^2=42$

Let the line $\mathrm{L}:\sqrt{2}x+y=\alpha$ pass through the point of the intersection $\mathrm{P}$ (in the first quadrant) of the circle $x^2+y^2=3$ and the parabola $x^2=2y$. Let the line $\mathrm{L}$ touch two circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y$-axis, then the square of the area of the triangle $\mathrm{P}{\mathrm{Q}}_1{\mathrm{Q}}_2$ is equal to ____.

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 $\mathrm{Q}$. Then, the length of $\mathrm{PQ}$ is :

1. A.$\frac{1}{2}$
2. B.1
3. C.$\frac{1}{\sqrt{2}}$
4. D.$\sqrt{2}$

If one of the diameters of the circle $x^2+y^2-10x+4y+13=0$ is a chord of another circle $\mathrm{C}$, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle $\mathrm{C}$ is :

1. A.4
2. B.$3\sqrt{2}$
3. C.6
4. D.$\sqrt{20}$

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$, meet the positive co-ordinate axes at the points $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to

1. A.12
2. B.20
3. C.24
4. D.18

If the circles $(x+1{)}^2+(y+2{)}^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then

1. A.$\frac{1}{2}<r<7$
2. B.$3<r<7$
3. C.$5<r<9$
4. D.$0<r<7$

Let $(5,\frac{a}{4})$ be the circumcenter of a triangle with vertices $A(a,-2),B(a,6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha +\beta +\gamma$ is

1. A.$60$
2. B.$62$
3. C.$53$
4. D.$30$

The area (in sq. units) of the part of the circle $x^2+y^2=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha }{2\beta }-\frac{65}{2}+\frac{\alpha }{\beta }{\sin }^{-1}\left(\frac{12}{13}\right)$, where $\alpha ,\beta$ are coprime numbers. Then $\alpha +\beta$ is equal to _______.

Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $(-\frac{22}{7},-4)$ and $(-\frac{1}{7},3)$ intersects the circle at only one point $P(\alpha ,\beta )$. Then, $17\beta -\alpha$ is equal to _______.

If the shortest distance of the parabola $y^2=4x$ from the centre of the circle $x^2+y^2-4x-16y+64=0$ is $\mathrm{d}$, then ${\mathrm{d}}^2$ is equal to :

1. A.16
2. B.24
3. C.20
4. D.36

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

1. A.$\frac{3}{13}$
2. B.$\frac{2}{13}$
3. C.$\frac{5}{13}$
4. D.$\frac{1}{13}$

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 distance from the origin is $4\sqrt{2}$, then $(\alpha +\beta {)}^2$ is equal to _______.

Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x=2\sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha ,\beta )$, then ${\alpha }^2-{\beta }^2$ is equal to :

1. A.$61$
2. B.$\frac{304}{5}$
3. C.$60$
4. D.$\frac{314}{5}$

Let $w=z\bar{z}+k_{1}z+k_{2}iz+\lambda (1+i),k_1,k_2\in \mathbb{R}$. Let $\mathrm{Re}(w)=0$ be the circle $C$ of radius 1 in the first quadrant touching the line $y=1$ and the $y$-axis. If the curve $\mathrm{Im}(w)=0$ intersects $C$ at A and B, then $30(AB{)}^2$ is equal to _________

Let the centre of a circle $\mathrm{C}$ be $(\alpha ,\beta )$ and its radius $r<8$. Let $3x+4y=24$ and $3x-4y=32$ be two tangents and $4x+3y=1$ be a normal to $\mathrm{C}$. Then $(\alpha -\beta +r)$ is equal to :

1. A.$7$
2. B.$9$
3. C.$5$
4. D.$6$

Let $\mathrm{C}$ be the circle in the complex plane with centre $z_0=\frac{1}{2}(1+3i)$ and radius $r=1$. Let $z_1=1+i$ and the complex number $z_2$ be outside the circle $C$ such that $|z_1-z_0||z_2-z_0|=1$. If $z_0,z_1$ and $z_2$ are collinear, then the smaller value of $|z_2{|}^2$ is equal to :

1. A.$\frac{3}{2}$
2. B.$\frac{5}{2}$
3. C.$\frac{13}{2}$
4. D.$\frac{7}{2}$

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 line $x+y=2$. Then $r_1^2+r_2^2-r_1{r}_2$ is equal to _________.

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 ${\text{E}}_k$. 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 :

1. A.2870
2. B.3210
3. C.3320
4. D.3080