Question

If $\alpha $ and $\beta$ are different complex numbers with $ | \beta | = 1$, then $\frac {\beta-\alpha}{1-\overline {\alpha} \beta}$ is equal to

• A. $\frac{1}{2}$
• B. $1$
• C. $\frac{1}{3}$
• D. $2$

Answer 1

The Correct Option is B

Since, \(|\beta|=1\)
\(\therefore|\beta|^{2}=\beta \bar{\beta}=1\)
\(\therefore\left|\frac{\beta-\alpha}{1-\bar{\alpha} \beta}\right| =\left|\frac{\beta-\alpha}{\beta \bar{\beta}-\bar{\alpha} \beta}\right|\)
\(=\frac{|\beta-\alpha|}{|\beta||\bar{\beta}-\bar{\alpha}|}\)
\(=\frac{|\beta-\alpha|}{1 \cdot|\overline{\beta-\alpha}|}\)
\(=1\)

Any integer that may be expressed as a+ib is referred to be a complex number. Complex numbers, such 9+3i and 7+8i, are an example. Here i = -1. This allows us to state that i² = 1. Therefore, we may use i = -1 for any equation that does not have a true solution.

A polynomial with two roots or one of degree two is referred to as a quadratic equation. A quadratic equation has the generic form y=ax²+bx+c. The real numbers a ≠ 0, b, and c are real numbers here. 

Mathematical ideas such as complex numbers and quadratic arithmetic deal with significant theories, concepts, and formulae. It combines the roots connected to a complex number set, known as complex roots, with line and quadratic measurements.

Real and imaginary categories are included in complex mathematical numbers. Real and imaginary numbers can be combined to form complex numbers. Typically, the real values are 1, 1998, and 12.38, but doubling the hypothetical numbers results in a negative number. A quadratic equation, on the other hand, is an algebraic mathematical equation that contains squares. Its name derives from the word "quad," which is a square. Likewise referred to as the "equation of a degree 2." The common quadratic equation is stated using complex numbers and quadratic arithmetic.

ax² + bx + c = 0