Prove that ##12≤OP≤13## in the problem involving complex numbers

[chwala]

Homework Statement

See attached.

Relevant Equations

Complex numbers

Find the question below; note that no solution is provided for this question.

My approach;
Find part of my sketch here;

* My diagram may not be accurate..i just noted that, $OP$ takes smallest value of $12$ when $|z+5|=|z-5|$ i.e at the end of its minor axis and greatest value $13$ at end of its major axis

We have been given,
$|z-5|+(z+5|=26$
Then $OP=|z|=|\dfrac{1}{2}((z+5)+(z-5))|$
$≤ \dfrac{1}{2}(|z+5|+|z-5|)=\dfrac{1}{2}×26=13$

Also,
$(|z+5|+|z-5|)^2=676$ and $(|z+5|-|z-5|)^2≥0$, adding this two gives us
$2|z+5|^2+2|z-5|^2≥676$
$⇒|z+5|^2+|z-5|^2≥338$
$(z+5)(z^*+5^*)+(z-5)(z^*-5^*)≥338$
$2zz^*+50≥338$
$2zz^*≥288$
$zz^*≥144$, which is $|z|^2≥144$
$⇒z≥12$, therefore $12≤OP≤13$

 

[PeroK]

Looks good.