Advertisements
Advertisements
प्रश्न
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Advertisements
उत्तर
\[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy . . . . (1)\]
\[ \Rightarrow \left[ \bar{\frac{\left( a^2 + 1 \right)^2}{2a - i}} \right] = \bar{{x + iy}}\]
\[ \Rightarrow \frac{\left( a^2 + 1 \right)^2}{2a + i} = x - iy . . . . (2)\]
\[\text { On multiplying (1) and (2), we get }\]
\[\frac{\left( a^2 + 1 \right)^2}{2a - i} \times \frac{\left( a^2 + 1 \right)^2}{2a + i} = \left( x + iy \right)\left( x - iy \right)\]
\[ \Rightarrow \frac{\left( a^2 + 1 \right)^4}{\left( 2a \right)^2 - i^2} = x^2 - i^2 y^2 \]
\[ \Rightarrow \frac{\left( a^2 + 1 \right)^4}{\left( 2a \right)^2 + 1} = x^2 + y^2\]
Hence,
\[x^2 + y^2 = \frac{\left( a^2 + 1 \right)^4}{4 a^2 + 1}\].
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Evaluate: `[i^18 + (1/i)^25]^3`
Evaluate the following:
(ii) i528
Evaluate the following:
\[\frac{1}{i^{58}}\]
Find the value of the following expression:
i30 + i80 + i120
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write 1 − i in polar form.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The principal value of the amplitude of (1 + i) is
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
Which of the following is correct for any two complex numbers z1 and z2?
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i93
Evaluate the following : i403
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
Show that 1 + i10 + i20 + i30 is a real number
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
