Problem 108 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\bar{b} z+b \bar{z}=c, b \neq 0\), be a line in the complex plane, where \(\bar{b}\) is the complex conjugate of \(b .\) If a point \(z_{1}\) is the reflection of a point \(z_{2}\) through the line, then \(\bar{z}_{1} b+z_{2} \bar{b}=\) (A) \(4 c\) (B) \(2 c\) (C) \(_{C}\) (D) None of these

Step-by-Step Solution

Answer

Option (B) \( 2c \).

Step 1: Express line equation

The given equation of the line is \( \bar{b}z + b\bar{z} = c \)."

Step 2: Reflect a point in a line

If a point \( z_1 \) is the reflection of \( z_2 \) across the line, the midpoint \( M \) between \( z_1 \) and \( z_2 \) should be on the line. Therefore, \( M = \frac{z_1 + z_2}{2} \) satisfies \( \bar{b}M + b\bar{M} = c \)."

Step 3: Substitute midpoint into line equation

Substitute \( M = \frac{z_1 + z_2}{2} \) into the line equation: \[ \bar{b} \left(\frac{z_1 + z_2}{2}\right) + b \left(\frac{\bar{z}_1 + \bar{z}_2}{2}\right) = c \]. Simplifying, we obtain \( \frac{\bar{b}(z_1 + z_2) + b(\bar{z}_1 + \bar{z}_2)}{2} = c \)."

Step 4: Solve for conjugate sums

Multiply through by 2 and rearrange terms to isolate \( \bar{z}_1 b + z_2 \bar{b} \):\[ \bar{b} z_1 + \bar{b} z_2 + b \bar{z}_1 + b \bar{z}_2 = 2c \].Since the conjugates reflect symmetrically, rearranging gives \( \bar{z}_1 b + z_2 \bar{b} = 2c \).

Key Concepts

Line Equation in Complex Numbers

Understanding line equations in the complex plane requires a shift from traditional 2D Cartesian coordinates to our friend, the complex number. A complex number, denoted by \( z = x + yi \), has a real part \( x \) and an imaginary part \( y \). The beauty of complex numbers is that they inherently hold both horizontal and vertical dimensions.
To describe a line in this plane, we leverage complex conjugates. The equation \( \bar{b}z + b\bar{z} = c \), where \( \bar{b} \) is the conjugate of complex \( b \), effectively behaves as a linear equation.
Here's how it works:

• Each term \( \bar{b}z \) and \( b\bar{z} \) contributes to forming what can be interpreted as a 'slope'.
• These terms use the complex conjugate, which flips the sign of the imaginary part, thereby serving the role of y-interchange.

As you manipulate these terms, think of them as translating and rotating the 'line' formed by points in the complex plane.

Reflection Point in Geometry

Reflecting a point across a line in the complex plane is an elegant process reminiscent of finding reflections in regular Euclidean geometry, but with a complex twist. When a point \( z_1 \) is the reflection of a point \( z_2 \) across a line, the midpoint between these two points lies exactly on the line.
The midpoint \( M \) is calculated using \( M = \frac{z_1 + z_2}{2} \). This midpoint ensures symmetrical distances from the line.
In terms of steps:

• Verify the midpoint is on the line by substituting \( M \) into the line equation.
• Ensure the equation \( \bar{b}M + b\bar{M} = c \) holds true to confirm symmetry.
• Use this midpoint relationship to express how \( z_1 \) and \( z_2 \) are linked through the line.

This setup allows the reflection properties to dictate how the elements interact, ultimately providing a solution through elegant symmetry.

Complex Conjugate Properties

The complex conjugate, notated as \( \bar{z} \) for a complex number \( z = x + yi \), has an intriguing property: it flips the imaginary part, converting \( yi \) to \(-yi \). This subtle transformation becomes quite powerful in geometric interpretations.
Here’s a quick breakdown:

• The conjugate maintains the modulus \( |z| \) of the complex number, effectively preserving its 'magnitude'.
• The conjugate operation reflects the complex number across the real axis—this reflection is crucial in determining the geometric positions of points.
• Properties like \( z\bar{z} = x^2 + y^2 \) offer tools for exploring and simplifying expressions involving reflection without losing generality.

By utilizing these properties, reflection and symmetry become simpler to handle, allowing us to draw connections between algebraic manipulations and geometric interpretations on the complex plane.