Circles · JEE Main previous year questions

A spherical iron ball of $10\text{ cm}$ radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{\text{ cm}}^3/\text{min}$. When the thickness of ice is $5\text{ cm}$, then the rate (in $\text{cm}/\text{min}$.) at which the thickness of ice decreases, is :

1. A.$\frac{1}{18\pi }$
2. B.$\frac{1}{36\pi }$
3. C.$\frac{1}{54\pi }$
4. D.$\frac{5}{6\pi }$

A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?

1. A.3x – 4y – 24 = 0
2. B.4x + 3y – 8 = 0
3. C.3x + 4y – 6 = 0
4. D.4x – 3y + 17 = 0

If a line, $y=mx+c$ is a tangent to the circle, $(x-3{)}^2+y^2=1$ and it is perpendicular to a line $L_1$, where $L_1$ is the tangent to the circle, $x^2+y^2=1$ at the point $\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$, then :

1. A.$c^2+6c+7=0$
2. B.$c^2-7c+6=0$
3. C.$c^2-6c+7=0$
4. D.$c^2+7c+6=0$

Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points A and B. The $(AB{)}^2$ is equal to :

1. A.$\frac{56}{5}$
2. B.$\frac{32}{5}$
3. C.$\frac{52}{5}$
4. D.$\frac{64}{5}$

If the angle of intersection at a point where the two circles with radii $5\mathrm{cm}$ and $12\mathrm{cm}$ intersect is $90^{\circ }$, then the length (in $\mathrm{cm}$) of their common chord is :

1. A.$\frac{13}{5}$
2. B.$\frac{60}{13}$
3. C.$\frac{120}{13}$
4. D.$\frac{13}{2}$

If the circles $x^2+y^2+5\mathrm{K}x+2y+\mathrm{K}=0$ and $2(x^2+y^2)+2\mathrm{K}x+3y-1=0$, $(\mathrm{K}\in \mathrm{R})$, intersect at the points P and Q, then the line $4x+5y-\mathrm{K}=0$ passes through P and Q, for :

1. A.exactly two values of K
2. B.no value of K
3. C.exactly one value of K
4. D.infinitely many values of K

The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, -3), then its radius is :

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

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of $50\mathrm{c}{\mathrm{m}}^3/min$. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :

1. A.$\frac{5}{6\pi }$
2. B.$\frac{1}{9\pi }$
3. C.$\frac{1}{36\pi }$
4. D.$\frac{1}{18\pi }$

The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the y-axis and lie in the first quadrant, is :

1. A.$x=\sqrt{1+2y},y\ge 0$
2. B.$y=\sqrt{1+2x},x\ge 0$
3. C.$y=\sqrt{1+4x},x\ge 0$
4. D.$x=\sqrt{1+4y},y\ge 0$

If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is :

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

A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :

1. A.72
2. B.84
3. C.56
4. D.98

The common tangent to the circles $x^2+y^2=4$ and $x^2+y^2+6x+8y-24=0$ also passes through the point :

1. A.$(6,-2)$
2. B.$(4,-2)$
3. C.$(-4,6)$
4. D.$(-6,4)$

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^2=4x$ at the point (1, 2) and the x-axis is :-

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

Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$, then its equation is :

1. A.$x{\log }_e|y|=2(x-1)$
2. B.$x^2{\log }_e|y|=-2(x-1)$
3. C.$x{\log }_e|y|=x-1$
4. D.$x{\log }_e|y|=-2(x-1)$

The tangent and the normal lines at the point $(\sqrt{3},1)$ to the circle $x^2+y^2=4$ and the x-axis form a triangle. The area of this triangle (in square units) is :

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

If a variable line, $3x+4y-\lambda =0$ is such that the two circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-2y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval :

1. A.$(23,31)$
2. B.$(2,17)$
3. C.$[13,23]$
4. D.$[12,21]$

Let $C_1$ and $C_2$ be the centres of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2-6x-6y+14=0$ respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $\mathrm{P}{\mathrm{C}}_1\mathrm{Q}{\mathrm{C}}_2$ is :

1. A.4
2. B.6
3. C.9
4. D.8

If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is :

1. A.$(x^2+y^2{)}^2=4R^2{x}^2{y}^2$
2. B.$(x^2+y^2)(x+y)=R^{2}xy$
3. C.$(x^2+y^2{)}^2=4R{x}^2{y}^2$
4. D.$(x^2+y^2{)}^3=4R^2{x}^2{y}^2$

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :

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

Three circles of radii $a,b,c$ $(a<b<c)$ touch each other externally. If they have x-axis as a common tangent, then :

1. A.a, b, c are in A.P.
2. B.$\sqrt{a},\sqrt{b},\sqrt{c}$ are in A.P
3. C.$\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}=\frac{1}{\sqrt{a}}$
4. D.$\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}$

Equation of a common tangent to the circle, $x^2+y^2-6x=0$ and the parabola, $y^2=4x$ is :

1. A.$2\sqrt{3}y=12x+1$
2. B.$\sqrt{3}y=x+3$
3. C.$2\sqrt{3}y=-x-12$
4. D.$\sqrt{3}y=3x+1$

If a circle C, whose radius is 3, touches externally the circle, $x^2+y^2+2x-4y-4=0$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

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

Let P be a point on the parabola, $x^2=4y$. If the distance of P from the center of the circle, $x^2+y^2+6x+8=0$ is minimum, then the equation of the tangent to the parabola at P, is :

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

A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $y-4x+3=0$, then its radius is equal to :

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

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is :

1. A.$\frac{3\sqrt{5}}{2}$
2. B.$\sqrt{10}$
3. C.$2\sqrt{10}$
4. D.$3\sqrt{\frac{5}{2}}$