Most Important Selected Qs for JEE AdvancedMathematicsComplex Number

Find the minimum value of the expression E=|z|^2+|z-3|^2+|z-6 i|^2 (where z=x+i y, x, y R ).

Correct answer

Step-by-step solution

Let z=x+i y aligned & E=z z +(z-3)( z -3)+(z-6 i)( z +6 i) &=3 z z -3(z+ z )+9+6(z- z ) i+36 (z+ z =2 x ; z- z =2 iy ) &=3 (x^2+y^2 )-6 x-12 y+45 &=3 [x^2+y^2-2 x-4 y+15 ] & E=3 [(x-1)^2+(y-2)^2+10 ] & hence E_ =30 when x=1 and y=2 i.e. z=1+2 i ? aligned

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