Question

A point ${}^{\prime }{z}^{\prime }$ moves on the curve $|z-4-3i|=2$ in an argand plane. The maximum and minimum values of $|z|$ are

• A. $2,1$
• B. $6,5$
• C. $4,3$
• D. $7,3$

Answer 1

The correct option is D $7,3$

Let $w=4+3i$.

We can write, $|z|=|(z-w)+w|$. Hence by triangle inequality ($|z_1+z_2|\le |z_1|+|z_2|$), we can write $|z|\le |z-w|+|w|$. It is given in the question that, $|z-w|=2$ and $|w|=\sqrt{4^2+3^2}=5$.

Putting the values, we get $|z|\le 7$.

Using another result of triangle inequality ($||z_1|-|z_2||\le |z_1+z_2|$), we can write $||z-w|-|w||\le |z-w+w|$.

Hence, we get $|z|\ge 3$. The minimum value is 3 and maximum value is 7.

Hence, (D) is the correct option