# If the complex number Z satisfies $|Z^2 - 9| + |Z^2| = 41$ then the true statements among the following are?

If the complex number Z satisfies $$|Z^2 - 9| + |Z^2| = 41$$ then the true statements among the following are ?

$$A)$$ $$|Z+3| + |Z-3| = 10$$
$$B)$$ $$|Z+3| + |Z-3| = 8$$
$$C)$$ Maximum value of $$|Z|$$ is $$5$$
$$D)$$ Maximum value of $$|Z|$$ is 6

(More than one option may be correct)

The inital equation in the question indicates the fact that the locus of $$Z^2$$ is an ellipse with foci at $$0$$ and $$9$$. But this doesn't help much.

If we put $$Z = x + iy$$ and simplify. We get an ellipse in $$Z$$. I get $$99y^2 + 63x^2 = 1600$$ (but I may be incorrect). How do I proceed from here to validate any of those options?

There seems to exist a much simpler way to solve this question.
All help will be appreciated.

Let $z=Z^2$ and the equation becomes $$|z-9|+|z|=41 \tag{1}$$ which is an ellipse whose foci are $(0,0)$ and $(9,0)$. Note that $|z|$ reaches it max when $z$ is real and using this, it is easy to see $\max|z|=25$ and hence $\max|Z|=5$. Therefore (C) is the answer.

You also can use the following method: Let $z=x+yi$ and $x=r\cos\theta,y=r\sin\theta$. Then (1) becomes $$\sqrt{r^2-18r\cos\theta+81} +r=41$$ which imples $$r^2-r^2-18r\cos\theta+81=(41-r)^2.$$ From this, one has $$r=\frac{1600}{2(41-9\cos\theta)} \tag{2}.$$ Clearly if $\cos\theta=1$, $r$ reaches the max $5$ or $\max|z|=25$. So $\max|Z|=5$.

• +1 Thanks for the answer!. I'm not sure how you reach to that conclusion (that max z is real) without making any more calculations? Commented Apr 30, 2018 at 15:25
• If you draw the graph of the ellipse $|z-9|+|z|=41$, you will see this conclusion. Commented Apr 30, 2018 at 15:38
• How do you get |Z| = 25 after that? I mean the only way I could do it is by writing the equation of the ellipse then solving. Commented Apr 30, 2018 at 15:39
• $\max|z|=25$ implies $\max|Z^2|=25$ or $\max|Z|=5$. Commented Apr 30, 2018 at 15:42
• "Therefore (C) is the answer." Note that the question says (More than one option may be correct) so all you can say is that C is one answer. Commented Apr 30, 2018 at 16:05

The other answer claims C is the only answer, but this is not true. To investigate A and B take the square of $(|z-3|+|z+3|)$ and see what you get:

$$(|z-3| + |z+3|)^2 = |z-3|^2 + |z+3|^2 + 2|z^2-9|$$ since $|z-3|\cdot |z+3| = |(z-3)(z+3)| = |z^2-9|$. Next $|z-3|^2 + |z+3|^2 = 2|z|^2 + 2\cdot 3^2$ by the parallelogram law so

$$(|z-3| + |z+3|)^2 = 2\left(|z^2| + |z^2-9|\right) + 18 = 2\cdot 41 + 18 = 10^2$$

This shows that A is also an answer.

• +1 Thanks for the answer. Now I'm really confused as to who's answer I should accept. Commented Apr 30, 2018 at 16:57
• @SmarthBansal I couldn't care less about that, but if you accept the other answer you should first ask the other answerer to remove his/her wrong claims (in the answer and comments). I tried to get this through, but got nowhere so therefore I added this answer. Commented Apr 30, 2018 at 17:00