For all complex numbers z 1 , z 2 satisfying | z 1 | = 12 and | z 2 - 3 - 4 i | = 5 , the minimum value of | z 1 - z 2 | is [2002]

Question

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

Smart Achievers · Accepted Answer

(2)

$| z_{1} |$ $= 12 \Rightarrow z_{1} lies on a circle with centre (0, 0) and radius 12 units.$

$And$ $| z_{2} - 3 - 4 i |$ $= 5 \Rightarrow z_{2} lies on a circle with centre (3, 4) and radius 5 units.$

$From figure, it is clear that$ $| z_{1} - z_{2} |$ $i.e., distance between z_{1} and z_{2} will be minimum when they lie at A and B respectively,$

$i.e., O, C, B, A are collinear as shown.$

$Then$ $| z_{1} - z_{2} |$ $= A B = O A - O B = 12 - 2 (5) = 2 .$

$As above is the minimum value, we must have$ $| z_{1} - z_{2} |$ $\geq 2 .$

Solution

Q.
For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

1 0

2 2

3 7

4 17

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Solution

Q. For all complex numbers z1,z2 satisfying |z1|=12 and |z2-3-4i|=5, the minimum value of |z1-z2| is [2002]

1 0

2 2

3 7

4 17

Want Us to Email you About Special Offers & Updates?

Q.
For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]