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If |z-2-i|=4 then find the minimum value of |3z-6+15i| plz reply fast as possible

If |z-2-i|=4 then find the minimum value of |3z-6+15i| plz reply fast as possible

Grade:12

2 Answers

kashish bhatia
25 Points
6 years ago
|z-2-i|=4z-2-i=+4,-4z=6+i,(-2+i)1.z=6+i|3(6+i)-6+15i|=|12+18i|2.z= -2+I|3(-2+i)-6+15i|=|12-18i|=MINIMUM
Samyak Jain
333 Points
5 years ago
|z – 2 – i| = 4  or  |z – (2 + i)| = 4  
Above equation represents a circle centred at (2,1) with radius 4 units.
\therefore Eqn of circle is (x – 2)2 + (y – 2)2 = 16            ................(1)
Minimum value of  |3z-6+15i| means minimum value of
3|z – (2 – 5i)|. 
First we’ll find the minimum value of |z – (2 – 5i)| & multiply it by 3.
It denotes least distance from a point on the circle & point (2, – 5).
(2, – 5) lies below the circle in coordinate axes system.
Line through center (2,1) & (2, – 5) is parallel to y-axis &
its eqn is x = 2          …................(2)
points of intersection of (1) & (2) are (2,5) & (2, – 3).
By geometry, (2, –3) is nearest to (2, –5) of all points on the circle. 
\therefore Distance between (2, –3) & (2, –5) is 2 units, i.e., 
minimum value of |z – (2 – 5i)| is 2 and hence 
the minimum value of |3z-6+15i| = 3|z – (2 – 5i)| = 3*2
                                                      = 6

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