Complex Optimization

Algebra Level 5

Maximize $\alpha$ if there is exists a complex number $z$ satisfying $\begin{aligned} &|z| =3\\ &|z-(\alpha-1)i-\alpha| \leq 3 \\ &|z-(\alpha+1)i+2\alpha| >3 \end{aligned}$

Details and Assumptions

$\alpha$ can only be real.

The answer is 4.713.

1 solution

Ayush Parasar
Dec 18, 2014

Geometrically we can see that the complex number z lies on a circle with center at the origin of the Argand plane. Now the first inequality given states that z should lie on or inside a circle with center at (a , a-1) and radius = 3. And the other inequality given states z should be outside a circle with center (2a , -a-1 ) .

Now from the first inequality , for "a" to be maximum , both the circles should be tangent to each other . Thus distance between (0, 0) and (a, a-1) should be equal to 3 + 3 i.e. 6. Hence,

$2a^{2} -2a - 35= 0$

On solving we get a(maximum) = 4.713

Note - the negative value of "a" should be ignored as (a , a-1) is in the first quadrant.

Complex Optimization

The answer is 4.713.

1 solution

0 pending reports