Question

If $\left| {z - 1} \right| + \left| {z + 3} \right| \leqslant 8$, then the range of values of $\left| {z - 4} \right|$
$\left( A \right)\left[ {1,7} \right]$
$\left( B \right)\left[ {1,8} \right]$
$\left( C \right)\left[ {1,9} \right]$
$\left( D \right)\left[ {2,5} \right]$

Answer 1

Hint – In this particular type of question use the concept that in a triangle if ${z_1}{\text{ and }}{z_2}$ are two complex numbers then modulus of the sum of the two complex number is always less than or equal to the sum of the individual modulus of the complex numbers so use this concept to reach the solution of the question.

Complete step by step solution:
As we know that in a triangle if ${z_1}{\text{ and }}{z_2}$ are two complex numbers then modulus of the sum of the two complex number is always less than or equal to the sum of the individual modulus of the complex numbers.
$\Rightarrow \left| {{z_1} + {z_2}} \right| \leqslant \left| {{z_1}} \right| + \left| {{z_2}} \right|$................. (1)
Given equation:
$\left| {z - 1} \right| + \left| {z + 3} \right| \leqslant 8$................... (2)
Let, ${z_1} = z - 1$ and ${z_2} = z + 3$
Now from equation (1) and (2) we have,
$\Rightarrow \left| {z - 1 + z + 3} \right| \leqslant 8$
$\Rightarrow \left| {2z + 2} \right| \leqslant 8$
2 is constant so it can come outside the modulus so we have,
$\Rightarrow 2\left| {z + 1} \right| \leqslant 8$
Now divide by 2 throughout we have,
$\Rightarrow \left| {z + 1} \right| \leqslant 4$..................... (3)
Now,
$\left| {z + 1} \right| \leqslant \left| z \right| + \left| 1 \right|$
$\Rightarrow \left| {z + 1} \right| \leqslant \left| z \right| + 1$..................... (4)
Now from equation (3) and (4) we have,
$\Rightarrow \left| z \right| + 1 = 4$
$\Rightarrow \left| z \right| = \left( {4 - 1} \right)$
$\Rightarrow \left| z \right| = 3$................ (5)
So from equation (5), value of z is 3
Now the maximum value of $\left| {{z_1} + {z_2}} \right| = \left| {\left| {{z_1}} \right| + \left| {{z_2}} \right|} \right|$
And the minimum value of $\left| {{z_1} + {z_2}} \right| = \left| {\left| {{z_1}} \right| - \left| {{z_2}} \right|} \right|$
So the maximum value of $\left| {z - 4} \right| = \left| {\left| z \right| + \left| 4 \right|} \right|$
And the minimum value of $\left| {z - 4} \right| = \left| {\left| z \right| - \left| 4 \right|} \right|$
In both of the cases we have to put the value of z.
Now substitute the values we have,
So the maximum value of $\left| {z - 4} \right| = \left| {3 + 4} \right| = 7$
And the minimum value of $\left| {z - 4} \right| = \left| {3 - 4} \right| = \left| { - 1} \right| = 1$
So the range of values of $\left| {z - 4} \right|$ is
$\Rightarrow 1 < \left| {z - 4} \right| < 7$
So the range is [1, 7].
So this is the required answer.
Hence option (A) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that always recall that the maximum value of $\left| {{z_1} + {z_2}} \right|$ is $\left| {\left| {{z_1}} \right| + \left| {{z_2}} \right|} \right|$ and the minimum value of $\left| {{z_1} + {z_2}} \right|$ is $\left| {\left| {{z_1}} \right| - \left| {{z_2}} \right|} \right|$ so first find out the values of ${z_1}{\text{ and }}{{\text{z}}_2}$ and substitute these values in the above written formulas and simplify we will get the required answer.