Question:
If $|z| \ge 3$, then the least value of $\left|z+\frac{1}{4}\right|$ is
If $|z| \ge 3$, then the least value of $\left|z+\frac{1}{4}\right|$ is
Updated On: Apr 15, 2024
- $\frac{11}{2}$
- $\frac{11}{4}$
- $3$
- $\frac{1}{4}$
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The Correct Option is B
Solution and Explanation
$\left|z+\frac{1}{4}\right|$
$= \left|z-\left(\frac{-1}{4}\right)\right| \ge \left|z\right|-\left|\frac{-1}{4}\right|$
$= \left|\left(-z\right)-\frac{1}{4}\right|\ge \left|3-\frac{1}{4}\right|= \frac{11}{4}$
Hence, $\left|z+\frac{1}{4}\right| \ge \frac{11}{4}$
$= \left|z-\left(\frac{-1}{4}\right)\right| \ge \left|z\right|-\left|\frac{-1}{4}\right|$
$= \left|\left(-z\right)-\frac{1}{4}\right|\ge \left|3-\frac{1}{4}\right|= \frac{11}{4}$
Hence, $\left|z+\frac{1}{4}\right| \ge \frac{11}{4}$
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.