Key Concepts and Formulas

• Modulus as Distance: For complex numbers $z_1$ and $z_2$, $|z_1 - z_2|$ represents the distance between the points corresponding to $z_1$ and $z_2$ in the complex plane.
• Perpendicular Bisector: The locus of points $z$ such that $|z - a| = |z - b|$ is the perpendicular bisector of the line segment joining $a$ and $b$.
• Complex Number Representation: A complex number $z$ can be written as $z = x + iy$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).

Step-by-Step Solution

Step 1: Understand the Problem

We are given that a complex number $z$ is equidistant from the complex numbers $1$, $-1$, and $i$. We need to find the number of such complex numbers. Geometrically, we are looking for a point equidistant from three given points in the complex plane. Since three non-collinear points uniquely define a circle, there can be at most one such point (the center of the circle).

Step 2: Set up the Equations

Let $z = x + iy$, where $x$ and $y$ are real numbers. The given condition is: $|z - 1| = |z + 1| = |z - i|$ This implies two separate equations:

1. $|z - 1| = |z + 1|$
2. $|z + 1| = |z - i|$

Step 3: Analyze the First Equation $|z - 1| = |z + 1|$

This means $z$ is equidistant from $1$ and $-1$.

• Step 3.1: Substitute $z = x + iy$ We replace $z$ with $x+iy$ in the equation: $|(x + iy) - 1| = |(x + iy) + 1|$ Group the real and imaginary parts: $|(x - 1) + iy| = |(x + 1) + iy|$

  • Why this step? By substituting $z=x+iy$, we convert the complex equation into an expression involving real variables $x$ and $y$. Grouping real and imaginary parts is essential before applying the modulus definition.

• Step 3.2: Apply the Modulus Definition Recall that for a complex number $w = a + bi$, its modulus is $|w| = \sqrt{a^2 + b^2}$. Applying this definition to both sides: $\sqrt{(x - 1)^2 + y^2} = \sqrt{(x + 1)^2 + y^2}$

  • Why this step? This eliminates the complex number notation and gives us a purely algebraic equation in terms of $x$ and $y$, which we can solve using standard algebraic techniques.

• Step 3.3: Solve for $x$ To eliminate the square roots, we square both sides of the equation: $(x - 1)^2 + y^2 = (x + 1)^2 + y^2$ Expand the squared terms: $x^2 - 2x + 1 + y^2 = x^2 + 2x + 1 + y^2$ Subtract $x^2 + y^2 + 1$ from both sides of the equation: $-2x = 2x$ Combine like terms: $4x = 0$ Divide by 4: $x = 0$

  • Why this step? This gives us the first crucial constraint on the complex number $z$. It tells us that the real part of $z$ must be zero.

Step 4: Analyze the Second Equation $|z + 1| = |z - i|$

This means $z$ is equidistant from $-1$ and $i$.

• Step 4.1: Substitute $z = x + iy$ Substitute $z = x+iy$ into the equation: $|(x + iy) + 1| = |(x + iy) - i|$ Group the real and imaginary parts: $|(x + 1) + iy| = |x + i(y - 1)|$

  • Why this step? Similar to the first condition, this transforms the complex equation into a form ready for the modulus definition.

• Step 4.2: Apply the Modulus Definition $\sqrt{(x + 1)^2 + y^2} = \sqrt{x^2 + (y - 1)^2}$

  • Why this step? This yields a second independent algebraic equation in $x$ and $y$. We need at least two such equations to uniquely determine $x$ and $y$.

• Step 4.3: Solve for $y$ Square both sides to remove the square roots: $(x + 1)^2 + y^2 = x^2 + (y - 1)^2$ Expand the squared terms: $x^2 + 2x + 1 + y^2 = x^2 + y^2 - 2y + 1$ Subtract $x^2 + y^2 + 1$ from both sides: $2x = -2y$ Divide by 2: $x = -y$

  • Why this step? This gives us the second constraint on $z$.

Step 5: Find the Unique Solution for $z$

Now we have a system of two linear equations for $x$ and $y$:

1. $x = 0$
2. $x = -y$

Substitute $x = 0$ into the second equation: $0 = -y$ $y = 0$

So, $x = 0$ and $y = 0$. Therefore, $z = x + iy = 0 + i(0) = 0$.

Step 6: Verification

We must verify that $z=0$ satisfies all three conditions implied by the original equation:

1. $|z-1| = |0-1| = |-1| = 1$
2. $|z+1| = |0+1| = |1| = 1$
3. $|z-i| = |0-i| = |-i| = 1$ Since $|z-1| = |z+1| = |z-i| = 1$, the solution $z=0$ is valid.

Common Mistakes & Tips

• Algebraic Errors: Be extremely careful when expanding squares and simplifying equations. Double-check each step to avoid sign errors or incorrect cancellations.
• Geometric Intuition: Always try to visualize the problem geometrically. The point equidistant from three given points is the circumcenter of the triangle they form. This can help you anticipate the solution and check your algebraic work.
• Verification: After finding a solution, verify that it satisfies all the given conditions. This is especially important when dealing with multiple equations or constraints.

Summary

We are looking for a complex number $z$ equidistant from $1, -1,$ and $i$. By setting $z = x + iy$ and using the modulus definition, we derived two equations: $x = 0$ and $x = -y$. Solving this system gave us $x = 0$ and $y = 0$, so $z = 0$. This solution satisfies the original condition. Therefore, there is only one such complex number.

Final Answer

The final answer is \boxed{1}, which corresponds to option (A).