Q:

equation of a circle whose center is a + i

I do not know how to get this one started. I have tried graphing the functions. This is a multiple choice question.

The equation of the circle whose centre is at a + i (where a is a real number) and intersecting two circles $|z| = 1$ and $|z - 1| = 4$ orthogonally is
a) $|z-7+i| = 7$
b) $|z-2+i| = 7$
c) $|z+7-i| = 7$
d) $|z+2+i| = 7$

so I attempted to draw these two circles but really I do if this is a valuable first step. 
$$|z|  = \sqrt{x^2 + y^2} = 1$$
$$|z-1| = \sqrt{(x-1)^2 + y^2} = 4$$ 

A:

Suppose the radius of the circle is $r$. Since the circle intersects the circle $|z|=1$ orthogonally by Pythagoras Theorem we must have $|a+i|^{2}=1^{2} + r^{2}$. Similarly we will have $|(a-1)+i|^{2}= 4^{2} + r^{2}$. Solving these two equations we will have $a=-7$ and $r^{2}=47$. So the equation of the circle should be $|z+7-i|=\sqrt{47}$.

