Question

If the complex number $z$ lies on the boundary of the circle of radius $3$ and centre at $(-4)$ then find the greatest value of $|z+i|$.

Answer 1

Given, the complex number $z$ lies on the boundary of the circle of radius $3$ and centre at $(-4)$ or $\left(\right(-4+0i)$.

Then the equation of the circle is $|z-(-4+0i)|=3$ or, $|z+4|=3$......(1).

Now,

$|z+1|=|(z+4)-3|\le |z+4|+|3|=3+3=6$ [ Using (1)].

or, $|z+1|\le 6$

So greatest value of $|z+1|$ is $6$.