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Tangent

The greatest ...

Question

The greatest and least value of $| z_{1} + z_{2} |$ If $z_{1} = 24 + 7 i a n d | z_{2} | = 6$ are respectively

A

31,25

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B

25,19

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C

31,19

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D

none

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Solution

The correct option is C 31,19
$| z_{2} | = 6$ resresents a circle of radius 6 centred at (0,0).
$z_{1} = 24 + 7 i$ represents a point A(24,7).
Let a diameter through A meet the circle in points B and C, then AB is least and AC is greatest distance of A from the circle.
Clearly $A O^{2} = 24^{2} + 7^{2} = 625$
$∴ A O = 25$
$∴$ Greatest distance $= A B = A O + B O = 25 - 6 = 31$
Least distance $= A B = A O - B O = 25 - 6 = 19$

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