Advertisements
Advertisements
प्रश्न
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
Advertisements
उत्तर
\[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2 = \left( a z_1 - b z_2 \right)\left( \bar{{a z_1 - b z_2}} \right) + \left( a z_2 + b z_1 \right)\left( \bar{{a z_2 + b z_1}} \right)\]
\[ = \left( a z_1 - b z_2 \right)\left( a \bar{{z_1}} - b \bar{{z_2}} \right) + \left( a z_2 + b z_1 \right)\left( a \bar{{z_2}} + b \bar{{z_1}} \right)\]
\[ = \left( a^2 z_1 \bar{{z_1}} - ab z_1 \bar{{z_2}} - ab z_2 \bar{{z_1}} + b^2 z_2 \bar{{z_2}} \right) + \left( a^2 z_2 \bar{{z_2}} + ab z_1 \bar{{z_2}} + ab z_2 \bar{{z_1}} + b^2 z_1 \bar{{z_1}} \right)\]
\[ = \left[ \left( a^2 + b^2 \right) z_1 \bar{{z_1}} + \left( a^2 + b^2 \right) z_2 \bar{{z_2}} \right]\]
\[ = \left[ \left( a^2 + b^2 \right)\left( z_1 \bar{{z_1}} + z_2 \bar{{z_2}} \right) \right]\]
\[ = \left[ \left( a^2 + b^2 \right)\left( \left| z_1 \right|^2 + \left| z_2 \right|^2 \right) \right]\]
Hence,
\[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2 = \left( a^2 + b^2 \right)\left( \left| z_1 \right|^2 + \left| z_2 \right|^2 \right)\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Evaluate: `[i^18 + (1/i)^25]^3`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
The value of \[(1 + i )^4 + (1 - i )^4\] is
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Find a and b if (a + ib) (1 + i) = 2 + i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + i)(1 − i)−1
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Evaluate the following : i35
Evaluate the following : i888
Show that 1 + i10 + i20 + i30 is a real number
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
The order relation is defined on the set of complex numbers.
Show that `(-1 + sqrt3 "i")^3` is a real number.
