\(If a + ib =\frac{(x+1)^2}{2x^2+1},\,prove\, that\, a^2 + b^2 =\frac{(x^2+1)^2}{(2x^2+1)}\)
Solution and Explanation
\(If a + ib =\frac{(x+1)^2}{2x^2+1}\)
\(=\frac{x^2+i^2+2xi}{2x^2+1}\)
\(=\frac{x^2-1+i2x}{2x^2+1}\)
\(=\frac{x^2-1}{2x^2+1}+1(\frac{2x}{2x^2+1})\)
on comparing real and imaginary parts, we obtain
\(a=\frac{x^2-1}{2x^2+1}\,and\,\,b=\frac{2x}{2x^2+1}\)
\(a^2+b^2=(\frac{x^2-1}{2x^2+1})+(\frac{2x}{2x^2+1})^2\)
\(=\frac{x^4+1-2x^2+4x^2}{(2x+1)^2}\)
\(\frac{x^2+1+2x^2}{(2x^2+1)^2}\)
\(=\frac{(x^2+1)^2}{(2x^2+1)^2}\)
\(=a^2+b^2=\frac{(x^2+1)^2}{(2x+1)^2}\)
Hence, proved.
Top Questions on Complex Numbers and Quadratic Equations
- Let $z=(1+i)(1+2i)(1+3i)\cdots(1+ni)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to
- JEE Main - 2026
- Mathematics
- Complex Numbers and Quadratic Equations
Let \( \alpha = \dfrac{-1 + i\sqrt{3}}{2} \) and \( \beta = \dfrac{-1 - i\sqrt{3}}{2} \), where \( i = \sqrt{-1} \). If
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of \( m \) is ___________.- JEE Main - 2026
- Mathematics
- Complex Numbers and Quadratic Equations
- The number of distinct real solutions of the equation \[ x|x + 4| + 3|x + 2| + 10 = 0 \] is
- JEE Main - 2026
- Mathematics
- Complex Numbers and Quadratic Equations
- Let $z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$ and $|z|^2 = 44200$. Find the value of $n$.
- JEE Main - 2026
- Mathematics
- Complex Numbers and Quadratic Equations
- Let $z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$ and $|z|^2 = 44200$. Find the value of $n$.
- JEE Main - 2026
- Mathematics
- Complex Numbers and Quadratic Equations
Questions Asked in CBSE Class XI exam
- Balance the following redox reactions by ion-electron method:
(a)MnO4- (aq) + I - (aq) → MnO2(s) + I2(s) (in basic medium)
(b) MnO4- (aq) + SO2(g) → Mn2+(aq) +HSO4- (aq) (in acidic solution)
(c) H2O2(aq)+Fe2+(aq) → Fe3+ (aq) + H2O (l) (in acidic solution)
(d) Cr2O72-+ SO2(g) → Cr3+ (aq) +SO42- (aq) (in acidic solution)- CBSE Class XI
- Oxidation Number
- Write the resonance structures for SO3 , NO2 and NO3-
- CBSE Class XI
- Kossel-Lewis Approach to Chemical Bonding
- At 700 K, equilibrium constant for the reaction:
\(H_2 (g) + I_2 (g) ⇋ 2HI (g)\)
is 54.8. If 0.5 mol L–1 of HI(g) is present at equilibrium at 700 K, what are the concentration of H2(g) and I2(g) assuming that we initially started with HI(g) and allowed it to reach equilibrium at 700 K?- CBSE Class XI
- Law Of Chemical Equilibrium And Equilibrium Constant
- Find the mean deviation about the mean for the data 4, 7, 8, 9, 10, 12, 13, 17.
- CBSE Class XI
- Statistics
Find the mean deviation about the mean for the data 38, 70, 48, 40, 42, 55, 63, 46, 54, 44.
- CBSE Class XI
- Statistics
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.