Question

If |z+7| ≤ 9, for z ∈ C, the greatest value of |Z+2| is __

|$Z_1$ + $Z_2$| ≤ |$Z_1$| + |$Z_2$|

Answer 1

Given |z+7| ≤9

⟹ |z+2|= |z+7-5|≤ |z+7|+ |5|

= |z+7| +5

≤9+5

⟹ |z+2| ≤14

So, the greatest value of |z+2|=14

Alternate method

We know that |$|Z_1|-|Z_2|$| ≤ $|Z_1|-|Z_2|$

Here take $|Z_1|$ = Z+2 $|Z_2|$ = 5

⇒ |(|z+2|- |5|)| ≤ |z+7| ≤9

⇒ |(|z+2|-5)| ≤9

⇒ -9 ≤ |z+2|-5 ≤9

-9 + 5 ≤ |z+2| ≤14

-4 ≤ |z+2|≤14

Least value of |z +2|= -4

Greatest value of |z + 2| = 14