# Among the complex numbers z which satisfy the conditions |z-25i| ≤ 15, find the number having the least +ve argument.?

Relevance

The complex number z satisfying the condition | z - 25 i ≤ 15 | are represented by

the points inside and on the circle. of radius 15 and center at point C (0,25)from the

figure, it is clear that the complex number z satisfying |z-25i≤15| and having least

positive argument correspond to the point P (x,y), which is the point of contact of a

ray coming from the origin and lying in the first quadrant to the above circle.

The positive argument of all other points within and on the circle are greater than

the argument of P. from the figure we have -OC = 25, CP = radius = 15 and angle

CPO = 90°Hence OP= √(OC² - CP²)= √(25² - 15² ) = 20

If angle PCO = θ , then PON = θ. Also cos θ = PC/OC = 3/5Therefore (x) = ON = OP

cos θ = 12 ,  and PN = 16  Hence P represents the complex number

z = x + iy = | 12 + 16 I

which is , therefore, the required value of z satisfying the condition. • Draw diagram --->

z = 10i

• Many ways. A fun & direct visual way is to consider it geometrically in the complex plane. Hint: If we asked the same question for |z| ≤ 15, what does this represent geometriclly?