Circles · JEE Main previous year questions

Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-\lambda }{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{6}}$ be ${\lambda }_1$ and ${\lambda }_2$. Then the radius of the circle passing through the points $(0,0)$, $({\lambda }_1,{\lambda }_2)$ and $({\lambda }_2,{\lambda }_1)$ is

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

Let the ellipse $3x^2+p{y}^2=4$ pass through the centre $C$ of the circle $x^2+y^2-2x-4y-11=0$ of radius $r$. Let $f_1,f_2$ be the focal distances of the point $C$ on the ellipse. Then $6f_1{f}_2-r$ is equal to

1. A.$78$
2. B.$68$
3. C.$70$
4. D.$74$

Let $r$ be the radius of the circle, which touches $x$-axis at point $(a,0),a<0$ and the parabola $y^2=9x$ at the point $(4,6)$. Then $r$ is equal to $\underline{}$.

Let $C$ be the circle $x^2+(y-1{)}^2=2$, $E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line $x+y=3$ touch the curves $C,E_1$ and $E_2$ at $P(x_1,y_1),Q(x_2,y_2)$ and $R(x_3,y_3)$ respectively. Given that $P$ is the mid point of the line segment $QR$ and $PQ=\frac{2\sqrt{2}}{3}$, the value of $9(x_1{y}_1+x_2{y}_2+x_3{y}_3)$ is equal to _________.

The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is

1. A.$\frac{7\sqrt{2}}{16}$
2. B.$\frac{7\sqrt{2}}{8}$
3. C.$\frac{7\sqrt{2}}{2}$
4. D.$\frac{7\sqrt{2}}{4}$

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

1. A.$34$
2. B.$32$
3. C.$35$
4. D.$33$

The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to _________ .

Let a circle C pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3x+2y+2=0$. Then the length of the chord, of the circle C, whose mid-point is $(1,2)$, is:

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

Let the equation of the circle, which touches $x$-axis at the point $(a,0),a>0$ and cuts off an intercept of length $b$ on $y$-axis be $x^2+y^2-\alpha x+\beta y+\gamma =0$. If the circle lies below $x$-axis, then the ordered pair $(2a,b^2)$ is equal to

1. A.$(\alpha ,{\beta }^2+4\gamma )$
2. B.$(\alpha ,{\beta }^2-4\gamma )$
3. C.$(\gamma ,{\beta }^2-4\alpha )$
4. D.$(\gamma ,{\beta }^2+4\alpha )$

Let circle $C$ be the image of $x^2+y^2-2x+4y-4=0$ in the line $2x-3y+5=0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha ,\beta )$, with $\beta <4$, lies on $C$ such that the length of the arc $AB$ is $(1/6{)}^{\text{th }}$ of the perimeter of $C$, then $\beta -\sqrt{3}\alpha$ is equal to

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

Let the shortest distance from $(a,0),a>0$, to the parabola $y^2=4x$ be $4$ . Then the equation of the circle passing through the point $(a,0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

1. A.$x^2+y^2-8x+7=0$
2. B.$x^2+y^2-6x+5=0$
3. C.$x^2+y^2-4x+3=0$
4. D.$x^2+y^2-10x+9=0$

The focus of the parabola $y^2=4x+16$ is the centre of the circle $C$ of radius $5$ . If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3x-y=0$ and $x+\lambda y=4$, are ${\lambda }_1$ and ${\lambda }_2$, ${\lambda }_1<{\lambda }_2$, then $12{\lambda }_1+29{\lambda }_2$ is equal to _________.

The area of the region, inside the circle $(x-2\sqrt{3}{)}^2+y^2=12$ and outside the parabola $y^2=2\sqrt{3}x$ is :

1. A.$3\pi -8$
2. B.$6\pi -8$
3. C.$3\pi +8$
4. D.$6\pi -16$

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $(\alpha ,\beta )$, then $3\beta -2\alpha$ is equal to :

1. A.10
2. B.12
3. C.14
4. D.15

Let the parabola $y=x^2+px-3$, meet the coordinate axes at the points $\mathrm{P},\mathrm{Q}$ and $\mathrm{R}$ . If the circle C with centre at $(-1,-1)$ passes through the points $P,Q$ and $R$, then the area of $△PQR$ is :

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

The parabola $y^2=4x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part is equal to :

1. A.$\frac{2}{3}+5{\sin }^{-1}\left(\frac{2}{\sqrt{5}}\right)$
2. B.$\frac{2}{3}+\sqrt{5}{\sin }^{-1}\left(\frac{2}{\sqrt{5}}\right)$
3. C.$\frac{1}{3}+5{\sin }^{-1}\left(\frac{2}{\sqrt{5}}\right)$
4. D.$\frac{1}{3}+\sqrt{5}{\sin }^{-1}\left(\frac{2}{\sqrt{5}}\right)$

Let a circle passing through $(2,0)$ have its centre at the point $(h,k)$. Let $(x_c,y_c)$ be the point of intersection of the lines $3x+5y=1$ and $(2+c)x+5c^{2}y=1$. If $h={\lim }_{c\to 1}{x}_c$ and $k={\lim }_{c\to 1}{y}_c$, then the equation of the circle is :

1. A.$5x^2+5y^2-4x-2y-12=0$
2. B.$25x^2+25y^2-20x+2y-60=0$
3. C.$25x^2+25y^2-2x+2y-60=0$
4. D.$5x^2+5y^2-4x+2y-12=0$

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 $(h,k)$, $4(h^2+k^2)$ is equal to ______.

Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $C$ are $(\alpha ,\beta )$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(h,k)$, then the value of $(\alpha +\beta )+2(h+k)$ equals

1. A.$81$
2. B.$15$
3. C.$51$
4. D.$5$

Consider the circle $C:x^2+y^2=4$ and the parabola $P:y^2=8x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha ,0)$ are bisected by the parabola $P$ is the interval $(p,q)$, then $(2q-p{)}^2$ is equal to ____.

Let the circles $C_1:(x-\alpha {)}^2+(y-\beta {)}^2=r_1^2$ and $C_2:(x-8{)}^2+(y-\frac{15}{2}{)}^2=r_2^2$ 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

1. A.$130$
2. B.$110$
3. C.$145$
4. D.$125$

The area of the region in the first quadrant inside the circle $x^2+y^2=8$ and outside the parabola $y^2=2x$ is equal to :

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

If the image of the point $(-4,5)$ in the line $x+2y=2$ lies on the circle $(x+4{)}^2+(y-3{)}^2=r^2$, then $r$ is equal to:

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

Let $\mathrm{S}$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $x$-axis. Let $\mathrm{C}$ be the circle with its centre at $\mathrm{A}(\sqrt{6},\sqrt{5})$ and passing through the point $\mathrm{S}$. If $\mathrm{O}$ is the origin and $\mathrm{SAB}$ is a diameter of $\mathrm{C}$, then the square of the area of the triangle $\mathrm{OSB}$ is equal to ____.

Suppose the solution of the differential equation $\frac{dy}{dx}=\frac{(2+\alpha )x-\beta y+2}{\beta x-2\alpha y-(\beta \gamma -4\alpha )}$ represents a circle passing through origin. Then the radius of this circle is :

1. A.$\sqrt{17}$
2. B.2
3. C.$\frac{\sqrt{17}}{2}$
4. D.$\frac{1}{2}$