Question

For positive constant $r,$ let $M$ be the set of complex numbers $z$ which satisfy $|z-4-3i|=r.$ Then which of the following statements is (are) CORRECT?

• A. If $r=3,$ then the minimum value of $|z|$ for complex number $z$ which belongs to $M$ is $2.$
• B. If $r=3,$ then the maximum value of $|z|$ for complex number $z$ which belongs to $M$ is $8.$
• C. If $r=5,$ then the complex number having least modulus which belongs to $M$ is $z=0$
• D. If $r=5,$ then the complex number having greatest modulus which belongs to $M$ is $z=8+6i$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A If $r=3,$ then the minimum value of $|z|$ for complex number $z$ which belongs to $M$ is $2.$
B If $r=3,$ then the maximum value of $|z|$ for complex number $z$ which belongs to $M$ is $8.$
C If $r=5,$ then the complex number having least modulus which belongs to $M$ is $z=0$
D If $r=5,$ then the complex number having greatest modulus which belongs to $M$ is $z=8+6i$

For $r=3,$ we have $|z-(4+3i)|=3$
Using property $||z_1|-|z_2||\le |z_1+z_2|\le |z_1|+|z_2|$
Clearly, $|z{|}_{max}=8$ and $|z{|}_{min}=2$

Also, for $r=5,$ we have $|z-(4+3i)|=5$
Clearly, complex number having least modulus is $z=0$
and greatest modulus is $z=8+6i.$