Circles · JEE Main previous year questions

Let ${\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_{15}$ be 15 points on a circle. The number of distinct triangles formed by points ${\mathrm{P}}_i,{\mathrm{P}}_j,{\mathrm{P}}_k$ such that $i+j+k\ne 15$, is :

1. A.12
2. B.419
3. C.443
4. D.455

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ and the circle $x^2+y^2=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to :

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

A circle C touches the line $x=2y$ at the point $(2,1)$ and intersects the circle $C_1:x^2+y^2+2y-5=0$ at two points P and Q such that PQ is a diameter of $C_1$. Then the diameter of C is :

1. A.$7\sqrt{5}$
2. B.$15$
3. C.$\sqrt{285}$
4. D.$4\sqrt{15}$

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

1. A.$\sqrt{53}$
2. B.9
3. C.8
4. D.$\sqrt{82}$

Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a^2>b^2$, passes through $\left(\sqrt{\frac{3}{2}},1\right)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus $F(\alpha ,0),\alpha >0$, of E and radius $\frac{2}{\sqrt{3}}$, intersects E at two points P and Q, then $P{Q}^2$ is equal to :

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

A spherical gas balloon of radius 16 meter subtends an angle $60^{\circ }$ at the eye of the observer A while the angle of elevation of its center from the eye of A is $75^{\circ }$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :

1. A.$8(2+2\sqrt{3}+\sqrt{2})$
2. B.$8(\sqrt{6}+\sqrt{2}+2)$
3. C.$8(\sqrt{2}+2+\sqrt{3})$
4. D.$8(\sqrt{6}-\sqrt{2}+2)$

The equation of a circle is $\mathrm{Re}(z^2)+2(\mathrm{Im}(z){)}^2+2\mathrm{Re}(z)=0$, where z = x + iy. A line which passes through the center of the given circle and the vertex of the parabola, $x^2-6x-y+13=0$, has y-intercept equal to ____.

Let the circle $S:36x^2+36y^2-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle S, then :

1. A.$\frac{25}{9}<C<\frac{13}{3}$
2. B.$100<C<165$
3. C.$81<C<156$
4. D.$100<C<156$

If in a triangle ABC, AB = 5 units, $\angle B={\cos }^{-1}\left(\frac{3}{5}\right)$ and radius of circumcircle of $\Delta \mathrm{ABC}$ is 5 units, then the area (in sq. units) of $\Delta \mathrm{ABC}$ is :

1. A.$10+6\sqrt{2}$
2. B.$8+2\sqrt{2}$
3. C.$6+8\sqrt{3}$
4. D.$4+2\sqrt{3}$

If the equation $a|z{|}^2+\overline{\alpha }z+\alpha \overline{z}+d=0$ represents a circle where $a,d$ are real constants then which of the following condition is correct?

1. A.$|\alpha {|}^2-ad\ne 0$
2. B.$|\alpha {|}^2-ad>0\text{ and }a\in \mathrm{R}-\{0\}$
3. C.$|\alpha {|}^2-ad\ge 0\text{ and }a\in \mathrm{R}$
4. D.$\alpha =0,a,d\in {\mathrm{R}}^{+}$

For the four circles M, N, O and P, following four equations are given : Circle M : $x^2+y^2=1$ Circle N : $x^2+y^2-2x=0$ Circle O : $x^2+y^2-2x-2y+1=0$ Circle P : $x^2+y^2-2y=0$ If the centre of circle M is joined with centre of the circle N, further center of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :

1. A.Rhombus
2. B.Square
3. C.Rectangle
4. D.Parallelogram

Choose the correct statement about two circles whose equations are given below : $x^2+y^2-10x-10y+41=0$ $x^2+y^2-22x-10y+137=0$

1. A.circles have same centre
2. B.circles have no meeting point
3. C.circles have only one meeting point
4. D.circles have two meeting points

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If R and r be the radius of circumcircle and incircle respectively of $\Delta \mathrm{ABC}$, then $(R+r)$ is equal to :

1. A.$7\sqrt{2}$
2. B.$\frac{9}{\sqrt{2}}$
3. C.$2\sqrt{2}$
4. D.$3\sqrt{2}$

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi }{3}$. If the radius of the circumcircle of $\Delta \mathrm{ABC}$ is 2, then the height of the pole is equal to :

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

Let ${\mathrm{S}}_1:x^2+y^2=9$ and ${\mathrm{S}}_2:(x-2{)}^2+y^2=1$. Then the locus of center of a variable circle S which touches ${\mathrm{S}}_1$ internally and ${\mathrm{S}}_2$ externally always passes through the points :

1. A.$\left(\frac{1}{2},\pm \frac{\sqrt{5}}{2}\right)$
2. B.$(1,\pm 2)$
3. C.$\left(2,\pm \frac{3}{2}\right)$
4. D.$(0,\pm \sqrt{3})$

The locus of the midpoints of the chord of the circle, $x^2+y^2=25$ which is tangent to the hyperbola, $\frac{x^2}{9}-\frac{y^2}{16}=1$ is :

1. A.$(x^2+y^2{)}^2-9x^2+16y^2=0$
2. B.$(x^2+y^2{)}^2-9x^2+144y^2=0$
3. C.$(x^2+y^2{)}^2-16x^2+9y^2=0$
4. D.$(x^2+y^2{)}^2-9x^2-16y^2=0$

Let ABCD be a square of side of unit length. Let a circle $C_1$ centered at A with unit radius is drawn. Another circle $C_2$ which touches $C_1$ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle $C_2$ meet the side AB at E. If the length of EB is $\alpha +\sqrt{3}\beta$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to _________.

1. A.$24\sqrt{2}$
2. B.$24\sqrt{3}$
3. C.$26\sqrt{2}$
4. D.$26\sqrt{3}$

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 $(PA{)}^2+(PB{)}^2$ have maximum value, then the points, P, A and B lie on :

1. A.a straight line
2. B.an ellipse
3. C.a parabola
4. D.a hyperbola

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, $x^2+y^2=1$ is a circle of radius r, then r is equal to :

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

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is :

1. A.An equilateral triangle having each of its side of length $\sqrt{3}r$.
2. B.An equilateral triangle of height $\frac{2r}{3}$.
3. C.A right angle triangle having two of its sides of length 2r and r.
4. D.An isosceles triangle with base equal to 2r.

A line is a common tangent to the circle $(x-3{)}^2+y^2=9$ and the parabola $y^2=4x$. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then $2(a+c)$ is equal to ______ .

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^2$ at the point $(2,4)$ is :

1. A.$\left(\frac{6}{5},\frac{53}{10}\right)$
2. B.$\left(\frac{3}{10},\frac{16}{5}\right)$
3. C.$\left(-\frac{53}{10},\frac{16}{5}\right)$
4. D.$\left(-\frac{16}{5},\frac{53}{10}\right)$

The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is _______ .

Let the latus rectum of the parabola $y^2=4x$ be the common chord to the circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ is :

1. A.8
2. B.12
3. C.$8\sqrt{5}$
4. D.$4\sqrt{5}$