Problem 128 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z\) satisfies \(|z+1|<|z-2|\), and \(\omega=3 z+2+i\), then (A) \(|\omega+1|<|\omega-8|\) (B) \(|\omega+1|<|\omega-7|\) (C) \(\omega+\bar{\omega}>7\) (D) \(|\omega+5|<|\omega-4|\)

Step-by-Step Solution

Answer

(D) \(|\omega+5| < |\omega-4|\) is the correct option.

Step 1: Understand the Inequality

The inequality \(|z+1|<|z-2|\) is given. This implies that the complex number \(z\) is closer to \(-1\) than it is to \(2\) on the complex plane. This describes a set of points on a plane often referred to as a region in a complex plane.

Step 2: Transform the Inequality into a Geometric Figure

To translate this inequality into a geometric figure, recall that the set of all points \(z\) such that \(|z-a|<|z-b|\) is the region closer to \(a\) (which is \(-1\) here) than to \(b\) (which is \(2\) here). This region is bounded by the perpendicular bisector of the line segment joining \(-1\) and \(2\). Hence, it represents half the plane to the left of this bisector.

Step 3: Substitute the Expression for \(\omega\)

Given \(\omega = 3z + 2 + i\), substitute this into the inequalities. To find whether an option is true, check which points through the transformation satisfy the proposed inequality of \(\omega\).

Step 4: Test Each Option

By testing each option through substitution, determine whether the inequality holds.

Step 5: Application to Option A: Inequality \(|\omega+1| < |\omega-8|\)

Substitute \(\omega = 3z + 2 + i\) into \(|\omega+1| < |\omega-8|\), checking if this holds based on the transformation rules. Calculate \(|\omega+1| = |3z + 3 + i|\) and \(|\omega-8| = |3z - 6 + i|\). Check inequality holds by comparing the distances on the complex plane.

Step 6: Application to Option B: Inequality \(|\omega+1| < |\omega-7|\)

Similarly verify the inequality by calculating based on \(\omega = 3z + 2 + i\). Check if \(|3z + 3 + i| < |3z - 5 + i|\) is true. After simplification, determine if it holds.

Step 7: Application to Option C: Inequality \(\omega+\bar{\omega}>7\)

Evaluate \(\omega+\bar{\omega} = (3z+2+i) + (3\bar{z}+2-i) = 6\text{Re}(z) + 4\). Check if \(6\text{Re}(z) + 4 > 7\), and determine the range of \(\text{Re}(z)\) which satisfies this inequality.

Step 8: Application to Option D: Inequality \(|\omega+5|<|\omega-4|\)

Substitute \(\omega = 3z + 2 + i\) back and check if \(|3z + 7 + i| < |3z - 2 + i|\). This option directly follows after confirming the distances following through inequality transformation.

Key Concepts

Geometric Interpretation of Complex Numbers

Complex numbers can be visually represented on the complex plane, a powerful tool that helps to understand their properties and behavior. In the complex plane, each complex number corresponds to a point with a specific "address," given by its real part, which can be considered analogous to the x-coordinate, and its imaginary part, analogous to the y-coordinate.

When considering the inequality \(|z+1| < |z-2|\), it’s crucial to understand that this means the complex number \(z\) is closer to \