Circles · JEE Main previous year questions

If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2+4y^2=36$ is $r$, then $12r^2$ is equal to :

1. A.$72$
2. B.$92$
3. C.$115$
4. D.$69$

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$, and $\overrightarrow{OQ}=\alpha \vec{u}+\beta \vec{v}$, then $\alpha ,{\beta }^2$ are the roots of the equation :

1. A.$x^2+x-2=0$
2. B.$3x^2+2x-1=0$
3. C.$3x^2-2x-1=0$
4. D.$x^2-x-2=0$

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 of the point, that divides the line segment AB in the ratio $2:3$, is a circle of radius :

1. A.$\frac{2}{3}\lambda$
2. B.$\frac{3}{5}\lambda$
3. C.$\frac{\sqrt{19}}{7}\lambda$
4. D.$\frac{\sqrt{19}}{5}\lambda$

If the points P and Q are respectively the circumcenter and the orthocentre of a $△\text{ABC}$, then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}$ is equal to :

1. A.$\overrightarrow{QP}$
2. B.$\overrightarrow{PQ}$
3. C.$2\overrightarrow{PQ}$
4. D.$2\overrightarrow{QP}$

Let a circle of radius 4 be concentric to the ellipse $15x^2+19y^2=285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle :

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{6}$
4. D.$\frac{\pi }{12}$

Consider 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 ___________.

Let $\mathrm{O}$ be the origin and $\mathrm{OP}$ and $\mathrm{OQ}$ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points $\mathrm{P}$ and $\mathrm{Q}$ on it. If the circumcircle of the triangle $\mathrm{OPQ}$ passes through the point $\left(\alpha ,\frac{1}{2}\right)$, then a value of $\alpha$ is :

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

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 point $P$ is above the line $AB$. The point $Q$ on the line segment $AB$ is the foot of perpendicular from $P$ on $AB$. If $PQ$ is equal to 11 units, then the value of $\alpha \beta$ is ________.

If the tangents at the points P and Q on the circle $x^2+y^2-2x+y=5$ meet at the point $R\left(\frac{9}{4},2\right)$, then the area of the triangle PQR is :

1. A.$\frac{13}{8}$
2. B.$\frac{5}{8}$
3. C.$\frac{5}{4}$
4. D.$\frac{13}{4}$

For $\alpha ,\beta ,z\in \mathbb{C}$ and $\lambda >1$, if $\sqrt{\lambda -1}$ is the radius of the circle $|z-\alpha {|}^2+|z-\beta {|}^2=2\lambda$, then $|\alpha -\beta |$ is equal to ________.

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha ,\beta )$ and $\gamma$, then $3(\alpha +\beta +\gamma )$ is equal to :

1. A.$12$
2. B.$10$
3. C.$11$
4. D.$9$

For a triangle $ABC$, the value of $\cos 2A+\cos 2B+\cos 2C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

1. A.$\overrightarrow{MA}\cdot \overrightarrow{MB}=-18$
2. B.$\sin 2A+\sin 2B+\sin 2C=\sin A+\sin B+\sin C$
3. C.perimeter of $\Delta ABC$ is $18\sqrt{3}$
4. D.area of $\Delta ABC$ is $\frac{27\sqrt{3}}{2}$

A wire of length $20\text{ m}$ is to be cut into two pieces. A piece of length $l_1$ is bent to make a square of area $A_1$ and the other piece of length $l_2$ is made into a circle of area $A_2$. If $2A_1+3A_2$ is minimum then $(\pi l_1):l_2$ is equal to :

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

Let a circle $C_1$ be obtained on rolling the circle $x^2+y^2-4x-6y+11=0$ upwards 4 units on the tangent $\mathrm{T}$ to it at the point $(3,2)$. Let $C_2$ be the image of $C_1$ in $\mathrm{T}$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is :

1. A.$2\left(2+\sqrt{2}\right)$
2. B.$4\left(1+\sqrt{2}\right)$
3. C.$3+2\sqrt{2}$
4. D.$2\left(1+\sqrt{2}\right)$

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 :

1. A.$(0,4]$
2. B.$(4,\infty )$
3. C.$(2,12]$
4. D.$(8,\infty )$

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 :

1. A.$6$
2. B.$5(1+\sqrt{2})$
3. C.$5$
4. D.$5\sqrt{2}$

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^2+y^2-3x+10y-15=0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to :

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

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 $S(\alpha ,\beta )$, then $4\alpha -7\beta$ is equal to ____.

A triangle is formed by the tangents at the point (2, 2) on the curves $y^2=2x$ and $x^2+y^2=4x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r^2$ is equal to ____.

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 $B(1,\beta )$. The image of the circle with AB as a diameter in the line $x+y+2=0$ is :

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

Let $z$ be a complex number such that $\left|\frac{z-2i}{z+i}\right|=2,z\ne -i$. Then $z$ lies on the circle of radius 2 and centre :

1. A.(0, $-2$)
2. B.(0, 0)
3. C.(0, 2)
4. D.(2, 0)

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 point S. If S lies on the line $2x-ky=1$, then k is equal to ____.

Let C be the largest circle centred at $(2,0)$ and inscribed in the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If $(1,\alpha )$ lies on C, then $10{\alpha }^2$ is equal to ____

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 of the circle $C_1$, is a circle of radius $r_i$. If ${\theta }_1=\frac{\pi }{3},{\theta }_3=\frac{2\pi }{3}$ and $r_1^2=r_2^2+r_3^2$, then ${\theta }_2$ is equal to :

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{3\pi }{4}$
4. D.$\frac{\pi }{6}$

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 $\mathrm{r}$ is the radius of circle $c_2$, then $\alpha +6{\mathrm{r}}^2$ is equal to ____.