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← IIT-JEE 2011 · Paper 1
MathematicsInteger+4Question 67 of 69

IIT-JEE 2011 · Paper 1, Mathematics question 67

If zz is any complex number satisfying ∣z−3−2i∣≤2|z-3-2i|\le2, then the minimum value of ∣2z−6+5i∣|2z-6+5i| is

Answer and worked solution

Official answer: 5

Rewrite. ∣2z−6+5i∣=2∣z−(3−52i)∣|2z-6+5i|=2\left|z-\left(3-\frac52i\right)\right|.

Distance. The disc has centre (3,2)(3,2), radius 2; the point (3,−52)\left(3,-\frac52\right) is 92\frac92 from the centre, so the nearest point of the disc is 92−2=52\frac92-2=\frac52 away.

Minimum 2⋅52=52\cdot\frac52=5.

Answer: 5

Topics:Geometry in the complex plane
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