Family of Circles

Q. The circle passing through the point $(-1,0)$ and touching the $y$-axis at $(0,2)$ also passes through the point:

1. $(-\frac{5}{2},2)$
2. $(3,5)$
3. $(-4,0)$
4. $(-\frac{3}{2},0)$

Q. 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

Q. The equation arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$ represents a circle with:

1. centre at $(0,0)$ and radius $\sqrt{2}$
2. centre at $(0,1)$ and radius $\sqrt{2}$
3. centre at $(0,-1)$ and radius $\sqrt{2}$
4. centre at $(0,1)$ and radius $2$

Q. Equation of the circle passing through the point $(1,1)$ and point of intersection of circles $x^2+y^2=6$ and $x^2+y^2-6x+8=0$ is

1. $x^2+y^2-6x+4=0$
2. $x^2+y^2-3x+1=0$
3. $x^2+y^2-4y+2=0$
4. $x^2+y^2-3y+1=0$

Q. The radius of the smallest circle which touches the straight-line $3x-y=6$ at $(1,-3)$ and also touches the line $y=x$ is
(Compute upto one place of decimal only)

1. $2$ units
2. $1.5$ units
3. $1.8$ units
4. $1.2$ units

Q. The equation(s) of the circle passing through the points of intersection of the circles $x^2+y^2-2x-4y-4=0,$ $x^2+y^2-10x-12y+40=0$ and having radius $4$ units is/are

1. $2x^2+2y^2-18x-22y+69=0$
2. $x^2+y^2+28x-22y+19=0$
3. $x^2+y^2-2y-15=0$
4. $x^2+y^2+2x-15=0$

Q. The radius of the circle touching the coordinate axes and the line $3x+4y=12$ may be

1. 6
2. 2
3. 3
4. 1

Q. Find the vector $\overrightarrow{a}$ of magnitude $5\sqrt{2}$, making an angle of $\frac{\pi }{4}$ with x-axis, $\frac{\pi }{2}$ with y-axis and an angle $\theta$ with z-axis.

Q. A variable straight line $AB$ divides the circumference of the circle $x^2+y^2=25$ in the ratio $1:2.$ If a tangent $CD$ is drawn to the smaller arc parallel to $AB,$ such that $ABCD$ is a rectangle $($as shown in the figure$),$ then locus of $C$ and $D$ is

1. $x^2+y^2=\frac{175}{4}$
2. $x^2+y^2=36$
3. $x^2+y^2=40$
4. $x^2+y^2=20$

Q. Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

1. $0<k<\frac{1}{2}$
2. $k\ge \frac{1}{2}$
3. $-\frac{1}{2}\le k\le \frac{1}{2}$
4. $k\le \frac{1}{2}$

Q. Find the equation of a family of circles touching the lines $x^2-y^2+2y-1=0$. (where $h$ and $k$ are parameters)

1. $x^2+y^2-2ky+\frac{k^2}{2}+k-\frac{1}{2}=0$
2. $x^2+y^2-2hx-2y+\frac{h^2}{2}+1=0$
3. $x^2+y^2-2hx+2y+\frac{h^2}{2}=0$
4. $x^2+y^2-2ky+\frac{k^2}{2}=0$

Q. $S$ is circle having center at $(0,a)$ and radius $b(b<a)$. $A$ is a variable circle centered at $(\alpha ,0)$ and touching circle $S$, meets the $x-$ axis at $M$ and $N$. If a point $P$ on the $y-$ axis such that $\angle MPN$ is independent from $\alpha$, then

1. $\angle MPN={\cos }^{-1}\left(\frac{b}{a}\right)$
2. $P\equiv (0,\sqrt{a^2-b^2})$
3. $\angle MPN=-{\cos }^{-1}\left(\frac{b}{a}\right)$
4. $P\equiv (0,-\sqrt{a^2-b^2})$

Q. Consider the circle $x^2+y^2-10x-6y+30=0.$ Let $O$ be the centre of the circle and tangents at $A(7,3)$ and $B(5,1)$ meet at $C.$ Let $S=0$ represents the family of circles passing through $A$ and $B.$ Then ,

1. The area of quadrilateral $OACB$ is $4$
2. The radical axis for the family of circles $S=0$ is $x+y=10$
3. The smallest possible circle of the family $S=0$ is $x^2+y^2-12x-4y+38=0$
4. The coordinates of point $C$ are $(7,1)$

Q. Consider the family of circles $x^2+y^2-2x-2\lambda y-8=0$ passing through two fixed points $A$ and $B$. Then the distance between the points $A$ and $B$ is
units

Q. If the curves $a{x}^2+4xy+2y^2+x+y+5=0$ and $a{x}^2+6xy+5y^2+2x+3y+8=0$ intersect at four concyclic points, then the value of $a$ is

1. $4$
2. $-4$
3. $-6$
4. $6$

Q. Five circles $C_1,C_2,C_3,C_4,C_5$ with radii $r_1,r_2,r_3,r_4,r_5$ respectively $(r_1<r_2<r_3<r_4<r_5)$ be such that $C_i$ and $C_i+1$ touch each other externally for all $i=1,2,3,4$. If all the five circles touch two straight lines $L_1$ and $L_2$ and $r_1=2$ and $r_5=32$, then $r_3$ is (units)

Q. If the curves $a{x}^2+4xy+2y^2+x+y+5=0$ and $a{x}^2+6xy+5y^2+2x+3y+8=0$ intersect at four concyclic points then the value of a is

1. 4
2. -6
3. -4
4. 6

Q. Equation of the circle passing through the point $(1,1)$ and point of intersection of circles $x^2+y^2=6$ and $x^2+y^2-6x+8=0$ is

1. $x^2+y^2-6x+4=0$
2. $x^2+y^2-3x+1=0$
3. $x^2+y^2-4y+2=0$
4. $x^2+y^2-3y+1=0$

Q. Find the angles between the lines
$x-\sqrt{3}y=1$

Q. If a circle passes through the point $(1,2)$ and cuts the circle $x^2+y^2=4$ orthogonally then equation of the locus of its centre is the straight line $2x+4y+9=0$.

Q. At the point $(2,3)$ on the curve $y=x^3-2x+1,$ the gradient of the curve increases $k$ times as fast as its abscissa, Then the value of $k$ is

Q. Conisder the circle $x^2+y^2-10x-6y+30=0$. Let $O$ be the centre of the circle and tangent at $A(7,3)$ and $B(5,1)$ meet at $C$. Let $S=0$ represents family of circles passing through $A$ and $B$, then

1. Area of quadrilateral $OACB=4$ sq. units
2. The radical axis for the family of circles $S=0$ is $x+y=10$
3. The smallest possible circle of the family $S=0$ is $x^2+y^2-12x-4y+38=0$
4. The coordinates of point $C$ are $(7,1)$

Q. 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. $3$
2. $2\sqrt{2}$
3. $2$
4. $3\sqrt{2}$

Q. Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

1. $0<k<\frac{1}{2}$
2. $k\ge \frac{1}{2}$
3. $-\frac{1}{2}\le k\le \frac{1}{2}$
4. $k\le \frac{1}{2}$

Q. The area common to the circles $r=a\sqrt{2}$ and $r=2a\cos \theta$ is:

1. $a^2\frac{\pi }{2}$
2. $a^2\pi$
3. $a^2(\pi +1)$
4. $a^2(\pi -1)$

Q. The range of value of $P$ such that the angle $\theta$ between the pair of tangents drawn from the point $\left(p.0\right)$ to the circle $x^2+y^2=1$ lies in $(\frac{\pi }{3},\pi )$, is

1. $(-3,-2)\cup (2,3)$
2. $(0,2)$
3. $(-2,-1)\cup (1,2)$
4. None of these

Q. Consider the family of circles $x^2+y^2-2x-2\lambda y-8=0$ passing through two fixed points $A$ and $B$. Then the distance between the points $A$ and $B$ is
units

Q.

Choose the correct answer in the following question:
The point on the curve $9y^2=x^3$, where the normal to the curve makes equal intercepts with the axes is

(a) $(4,\pm \frac{8}{3})$ (b) $(4,\frac{-8}{3})$
(c) $(4,\pm \frac{3}{8})$ (d) $(\pm 4,\frac{3}{8})$

Q. 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(in units) is

1. $2$
2. $3\sqrt{2}$
3. $3$
4. $2\sqrt{2}$

Q. Find the equation of the circle passing through the points of intersection of the circles $x^2+y^2-2x-4y-4=0$ and $x^2+y^2-10x-12y+40=0$ and whose radius is 4