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

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.

• "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"? – user10354138 Jul 26 '20 at 18:15

## 4 Answers

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?

• I would call those two points $i = (0,1)$ and $-i = (0,-1)$. – Jeppe Stig Nielsen Jul 27 '20 at 5:58
• I love how intuitively this explains why the equation is an ellipse. – HEGX64 Jul 27 '20 at 6:46

$$|z+i|+|z−i|=8$$ 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 $$z= \pm 4i$$.

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 $$x = 0, y = \pm 4 \implies z = \pm 4i.$$

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.