Question

Let circles $C_1$ and $C_2$ on Argand 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

The correct option is B $\frac{3}{10}$

$|z+1|=3|z-2|=7$

$(x+1{)}^2+y^2=(3{)}^2(x-2{)}^2+y^2=(7{)}^2$

Let $z_0=x+iy\to$ the centre of circle

$c_{1}c=r_1+r$

$c_{2}c=r_2-r$

$c_{1}c+c_{2}c=r_1+r_2$

$=3+7=10=2a$

$a=5$

$c_1{c}_2=2ac=3$

$c=3/10$