**LONG ANSWER TYPE QUESTIONS** 1) If $\alpha$ and $\beta$ are different complex numbers with $|\beta| = 1$, then find $\left|\frac{\beta - \alpha}{1 - \bar{\alpha}\beta}\right|$ 2) If $|a + ib| = 1$ then prove that $\frac{(1 + b + ia)}{(1 + b - ia)} = b + ia$ 3) If $z = x + iy$ and the real part of $\frac{\bar{z} + 2}{\bar{z} - 1}$ is $4$, show that $x^2 + y^2 - 3x + 2 = 0$

Step 1: Express $\bar{z}$ in terms of $x$ and $y$

Since $z = x + iy$, the complex conjugate $\bar{z}$ is obtained by reversing the sign of the imaginary part, giving $\bar{z} = x - iy$. We substitute this into the given expression $\frac{\bar{z} + 2}{\bar{z} - 1}$.

Step 2: Substitute and realize the denominator

Grouping the real and imaginary parts of both numerator and denominator gives $\frac{(x + 2) - iy}{(x - 1) - iy}$. To make the denominator a real number, we multiply both numerator and denominator by the conjugate of the denominator, which is $(x - 1) + iy$.

Step 3: Expand and simplify the fraction

Expanding the product in the numerator gives real part $(x + 2)(x - 1) - (-y^2) = x^2 + x - 2 + y^2$, and imaginary part $y(x + 2) - y(x - 1) = 3y$. The denominator simplifies to $(x - 1)^2 + y^2$.

Step 4: Equate the real part to $4$

The problem states that the real part of the expression is equal to 4. We set the real part equal to 4 and cross-multiply to form a polynomial equation.

Step 5: Cross-multiply and derive the required equation

Expanding the right side yields $4x^2 - 8x + 4 + 4y^2$. Subtracting $x^2 + y^2 + x - 2$ from both sides gives $3x^2 + 3y^2 - 9x + 6 = 0$. Dividing the entire equation by 3 gives the desired result $x^2 + y^2 - 3x + 2 = 0$.

More from this chapter

• Find all values of (1+i)^(1-i) in the form a + bi.
• Find all real solutions to the equation: √(x + 3 - 4√(x - 1)) + √(x + 8 - 6√(x - 1)) = 1.
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• MODIFIED: Given that variable_91, variable_91, variable_91 are roots of x³ - 3x² + 4 = 0, find the value of variable_91⁴ + variable_91⁴ + variable_91⁴.
• If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (-4,0), find the greatest and least values of |z+1|.
• Find the value of 2 x^(4)+5 x³+7 x²-x+41, when x=-2-√3 i
• Express (-√3+√(-2))(2 √3-i) in the form of a+i b
• Find the value of P such that the difference of the roots of the equation x²-P x+8=0 is 2.

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