Question

Let circles $C_1$ and $C_2$ an Argand plane be given by $|z+1|=3$ and $|z-2|=7$ respectively. If a variable circle $|z-z_0|=r$ be inside circle $C_2$ such that it touches $C_1$ externally and $C_2$ internally then locus of $z_0$ describes a conic $E$ whose eccentricity is equal to

• A. $\frac{1}{10}$
• B. $\frac{3}{10}$
• C. $\frac{5}{10}$
• D. $\frac{7}{10}$

Answer 1

Answer: (B)

$\frac{3}{10}$

We have $C_1:{(x+1)}^2+y^2=9$

$C_2:{(x-2)}^2+y^2=49$

Now $C{C}_1=r+r_1$

and $C{C}_2=r_2-r$

$\Rightarrow C{C}_1+C{C}_2=r_1+r_2$

Locus of C is an ellipse with focus at

$C_1$ and $C_2$

Now $r_1+r_2=2a=\mathrm{10....}\left(1\right)$

and $d_{C_1{C}_2}\left(focallength\right)=2ae=\mathrm{3....}\left(2\right)$

(1) and (2) $\Rightarrow$ eccentricity $e$ is $\frac{3}{10}$