Question

Given below are Matching type questions,with two columns(each having some items) each.Each item of Column$I$ has to be matched with the items of Column$II$, by encircling the correct match(es).
Note:An item of column$I$ can be matched with more than one item of column$II$.All the items of column$II$ have to be matched.
Observe the following columns:

Answer 1

$\left(A\right)C_1\equiv (1,3),r_1=\sqrt{1+9-9}=1$
and $C_2\equiv (-3,1),r_2=\sqrt{9+1-1}=3$
Now, $\left|C_1{C}_2\right|=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}>r_1+r_2$
Hence, circles neither cuts nor touch each other.
$\therefore$ Point of intersection$=0$
$\Rightarrow \lambda =0$
and $\mu =4$ (i.e., $2$ direct common tangents and 2 transverse common tangents)
$\therefore \lambda +\mu =4$
$\mu -\lambda =4$
${\lambda }^{\mu }+{\mu }^{\lambda }=1$
Options $(3,4)$ are correct
$\left(B\right)C_1\equiv (3,0),r_1=3$
and $C_2\equiv (-1,0).r_2=1$
Now, $C_1{C}_2=4=r_1+r_2$
Hence, circles touch each other externally.
$\therefore$ Point of intersection is one
$\therefore \lambda =1$ and $\mu =3$
$\therefore \lambda +\mu =4$
$\mu -\lambda =2$
and ${\lambda }^{\mu }+{\mu }^{\lambda }=1^3+3^1=4$
$\left(C\right)C_1\equiv (1,2),r_1=\sqrt{1+4-1}=2$
and $C_2\equiv (2,1),r_2=\sqrt{4+1-1}=2$
$\therefore \left|C_1{C}_2\right|=\sqrt{2}$
$\because |r_1-r_2|<\left|c_1{c}_2\right|<r_1+r_2$
Hence, circles cuts at two points.
$\therefore$ Point of intersection is two
$\therefore \lambda =2,\mu =2;$
$\lambda +\mu =4$
$\mu -\lambda =0$
${\lambda }^{\mu }+{\mu }^{\lambda }=2^2+2^2=8$