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If $z$ is a complex number satisfying the equation $|z+i|+|z−i|=8 $ then maximum value of $|z|$ is ?

I took $z$ as a point p on the graph and drew lines connecting it to $i$ and $-i$. I assumed $z=x+iy$. Therefore x and y should be maximum. If x and y are maximum, the triangle by i,-i and z has maximum area i.e. height is maximum. Taking i and -i as the base, max height come out when z is on the x axis at distance √15 from origin. But the answer is 4. Please solve it.

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    $\begingroup$ "Therefore $x$ and $y$ should be maximum" --- what does that actually mean? Note that $\lvert z-i\rvert+\lvert z+i\rvert=8$ is an ellipse whose axes are the real and imaginary axes, so where is this point "$x$ and $y$ are maximum"? $\endgroup$ – user10354138 Jul 26 '20 at 18:15
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What do you understand by $|z+i|+|z-i|=8$ ?

It's basically the sum of distances of a point from the points $(0,i)$ and the point $(0,-i)$ on the imaginary axis. The sum is a constant. What does that mean?

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  • $\begingroup$ I would call those two points $i = (0,1)$ and $-i = (0,-1)$. $\endgroup$ – Jeppe Stig Nielsen Jul 27 '20 at 5:58
  • $\begingroup$ I love how intuitively this explains why the equation is an ellipse. $\endgroup$ – HEGX64 Jul 27 '20 at 6:46
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$|z+i|+|z−i|=8$ is an ellipse with foci at $z=i$ and $z=-i$. The points with maximal modulus are at the end of the major axis, that is $z= \pm 4i$.

Or with pure arithmetic: $$ 2|z| = |(z-i) + (z+i)| \le |z+i|+|z−i|=8 \\ \implies |z| \le 4 $$ with equality for $z= \pm 4i$.

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Apply Minkowski Inequality:

$8 = |z-i| + |z+i| = \sqrt{x^2+(y-1)^2} + \sqrt{x^2+ (y+1)^2} \ge \sqrt{(x+x)^2 + ((y-1)+(y+1))^2} = \sqrt{(2x)^2 + (2y)^2} = 2\sqrt{x^2+y^2} = 2|z| \implies |z| \le 4 \implies |z|_{\text{max}} = 4$ , when $x = 0, y = \pm 4 \implies z = \pm 4i.$

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Note that $$|Z_1+Z_2| \le |Z_1|+ |Z_2|$$ So $$8=|z+i|+|z-i| \ge |z+i+z-i| \implies 8 \ge |2z| \implies |z| \le 4.$$ Also geometrically $|z+i|+|z-i|=2a$ represents an ellispes whosesemi major axis is $a=4$ this also gives the maximum possible value of $|z|$ (maximum distance between origin and any point on ellipse).

OP should realize that complex numbers do not admit max, min or inequality. It $|z|$ that is real and has max.

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