question_answer

Let circles \[{{C}_{1}}\] and \[{{C}_{2}}\] on 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 is 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)  \[\frac{7}{10}\]

Correct Answer: B

Solution :

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       \[C{{C}_{2}}\,={{r}_{2}}-r\] \[\Rightarrow \,\,C{{C}_{1}}+C{{C}_{2}}\,+{{r}_{1}}+{{r}_{2}}\] \[\therefore \] locus of C is an ellipse with focus at \[{{C}_{1}}\] and \[{{C}_{2}}\] Now \[{{r}_{1}}+{{r}_{2}}\,=2a=10\]                          ? and \[{{d}_{{{c}_{1}}{{c}_{2}}}}\] (focal length)\[=2ae=3\]      ? \[\therefore \]  and  \[\Rightarrow \,\,\] eccentricity ?e? is \[\frac{3}{10}\]