JEE MAIN - Mathematics (2021 - 1st September Evening Shift - No. 17)
If for the complex numbers z satisfying | z $$-$$ 2 $$-$$ 2i | $$\le$$ 1, the maximum value of | 3iz + 6 | is attained at a + ib, then a + b is equal to ______________.
Answer
5
Explanation
| z $$-$$ 2 $$-$$ 2i | $$\le$$ 1
| x + iy $$-$$ 2 $$-$$ 2i | $$\le$$ 1
|(x $$-$$ 2) + i(y $$-$$ 2)| $$\le$$ 1
(x $$-$$ 2)2 + (y $$-$$ 2)2 $$\le$$ 1
| 3iz + 6 |max at a + ib
$$\left| {3i} \right|\left| {z + {6 \over {3i}}} \right|$$
$$3{\left| {z - 2i} \right|_{\max }}$$
_1st_September_Evening_Shift_en_17_1.png)
From figure maximum distance at 3 + 2i
a + ib = 3 + 2i = a + b = 3 + 2 = 5 Ans.
| x + iy $$-$$ 2 $$-$$ 2i | $$\le$$ 1
|(x $$-$$ 2) + i(y $$-$$ 2)| $$\le$$ 1
(x $$-$$ 2)2 + (y $$-$$ 2)2 $$\le$$ 1
| 3iz + 6 |max at a + ib
$$\left| {3i} \right|\left| {z + {6 \over {3i}}} \right|$$
$$3{\left| {z - 2i} \right|_{\max }}$$
From figure maximum distance at 3 + 2i
a + ib = 3 + 2i = a + b = 3 + 2 = 5 Ans.
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