Family of Circles

Q. Each of the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2+2x-2y+1=0$ touches internally a circle of radius $2$. The equation of circles touching all the three circles, is

1. $x^2+{(y-\frac{7}{3})}^2=\frac{4}{9}$
2. $x^2+{(y+\frac{1}{3})}^2=\frac{4}{9}$
3. $x^2+{(y+\frac{7}{3})}^2=\frac{4}{9}$
4. $x^2+{(y-\frac{1}{3})}^2=\frac{4}{9}$

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 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 centre of the circle passing through $(0,0)$ and $(1,0)$ and touching the circle $x^2+y^2=9$ can be

1. $(\frac{1}{2},\frac{1}{2})$

2. $(\frac{1}{2},-\sqrt{2})$

3. $(\frac{3}{2},\frac{1}{2})$

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

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

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

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. 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. The equation of the circle through the points of intersection of $x^2+y^2-1=0$, $x^2+y^2-2x-4y+1=0$ and touching the line $x+2y=0$, is

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

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. $P\equiv (0,-\sqrt{a^2-b^2})$