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

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  • 7 months ago
    Favourite answer

    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.

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  • alex
    Lv 7
    7 months ago

    Draw diagram --->

    z = 10i            

  • rotchm
    Lv 7
    7 months ago

    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? 

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