Problem 208 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of complex numbers \(z\) such that \(|z-1|\) \(=|z+1|=|z-i|\) equals \(\quad[2010]\) (A) 1 (B) 2 (C) \(\infty\) (D) 0

Step-by-Step Solution

Answer

B: The number of such complex numbers is 1.

Step 1: Understand the geometric condition

The conditions \(|z-1| = |z+1|\) and \(|z-1| = |z-i|\) represent geometric distances in the complex plane. The equation \(|z-1| = |z+1|\) implies that \(z\) is equidistant from \(1\) and \(-1\), which means \(z\) lies on the perpendicular bisector of the line segment joining \(1\) and \(-1\). This perpendicular bisector is the imaginary axis (\(x = 0\)).

Step 2: Analyze second condition geometrically

The equation \(|z-1| = |z-i|\) suggests that \(z\) is equidistant from \(1\) and \(i\). The perpendicular bisector of the line joining \(1\) and \(i\) can be found to have the equation \(y = x - 1\), which is a line.

Step 3: Find intersection points

The solutions require solving for the intersections of the imaginary axis \(x = 0\) and the line \(y = x - 1\). Substitute \(x = 0\) into \(y = x - 1\) to find the y-coordinate of intersection: \(y = 0 - 1 = -1\). Thus the intersection point is \((0, -1)\).

Step 4: Count the solutions

The intersection point \((0, -1)\) corresponds to the complex number \(z = -i\). Since this intersection provides a single solution, there is exactly one complex number satisfying all given conditions.

Key Concepts

Geometric Interpretation

In the realm of complex numbers, geometric interpretation is a powerful tool in understanding equations and expressions. A complex number, denoted as \(z = x + yi\), can be visualized as a point \((x, y)\) on the complex plane. The real part \(x\) represents the horizontal displacement, while the imaginary part \(y\) represents the vertical displacement.
When we consider the equation \(|z - 1| = |z + 1| = |z - i|\), we are essentially looking at distances from the complex number \(z\) to specific points on this plane: \(1\), \(-1\), and \(i\). Each absolute value equation here signifies that \(z\) is equidistant from two given points, giving rise to a geometric locus.

• For \(|z - 1| = |z + 1|\), the locus is the imaginary axis, because a point \(z\) is equidistant from \(1\) and \(-1\) when it lies on the line perpendicular to the segment joining these points and passing through the midpoint of this segment.
• Similarly, \(|z - 1| = |z - i|\) leads us to another geometric locus, a line which can be calculated to have the equation \(y = x - 1\).

These geometric interpretations simplify the visualization of the problem and aid in understanding how to solve it.

Perpendicular Bisector

A perpendicular bisector is a line that divides a line segment into two equal lengths at a 90-degree angle. In the complex plane, it helps in easily visualizing equidistant conditions. When you have an equidistance equation like \(|z-1| = |z+1|\), it implies that the complex number \(z\) must lie on the perpendicular bisector of the line segment joining the points \(1\) and \(-1\).

• The points \(1\) and \(-1\) are purely real numbers on the complex plane. Their geometric relationship simplifies, as the perpendicular bisector of the segment connecting these points is the imaginary axis, that is \(x = 0\). This axis itself supports understanding of the zeros of the real part of \(z\).
• Another example is the perpendicular bisector of the line segment between the points \(1\) and \(i\). This bisector can be calculated to have the formula \(y = x - 1\), symbolizing another line. By realizing these bisectors, we ascertain where solutions may be found by finding their intersection if any.

These bisectors give us a visual aid in determining equidistance for each constraint.

Intersection of Lines

Finding the intersection of lines is crucial to solving complex equations involving multiple equidistance conditions. By interpreting each condition as defining a line or geometric place, the solution requires determining the point(s) where these lines intersect.
In our case, to solve \(|z-1| = |z+1| = |z-i|\), we examine the intersection of:

• The imaginary axis \(x = 0\), given by the condition \(|z-1| = |z+1|\).
• The line \(y = x - 1\), derived from the condition \(|z-1| = |z-i|\).

To find their intersection, substitute \(x = 0\) into the equation of the second line, \(y = x - 1\), to determine \(y\). We find \(y = 0 - 1 = -1\). Hence, the intersection occurs at the point \((0, -1)\), representing the complex number \(z = -i\).
This process demonstrates the method of solving the problem by translating it into intersecting line constructs that fulfill both equidistance criteria. The result is the singular solution we sought, where the intersection gives an exact location, confirming that \(z = -i\) is indeed a valid solution.