Advertisements
Advertisements
Question
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Advertisements
Solution
\[ \left( x + iy \right)\left( 2 - 3i \right) = 4 + i\]
\[2x - 3ix + 2iy - 3 i^2 y = 4 + i\]
\[2x + 3y + i\left( - 3x + 2y \right) = 4 + i\]
\[\text{Comparing both the sides:} \]
\[2x + 3y = 4 . . . . (1) \]
\[ - 3x + 2y = 1 . . . . (2)\]
\[\text { Multiplying equation (1) by 3 and equation (2) by 2 }: \]
\[ 6x + 9y = 12 . . . (3)\]
\[ - 6x + 4y = 2 . . . (4)\]
\[\text { Adding equations (3) and (4) }: \]
\[13y = 14\]
\[y = \frac{14}{13}\]
\[\text { Substituting the value of y in equation (1):} \]
\[2x + 3 \times \frac{14}{13} = 4\]
\[ \Rightarrow 2x = 4 - \frac{42}{13}\]
\[ \Rightarrow 2x = \frac{10}{13}\]
\[ \Rightarrow x = \frac{5}{13}\]
\[ \therefore x = \frac{5}{13}\text { and } y = \frac{14}{13} \]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Show that 1 + i10 + i20 + i30 is a real number.
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 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`
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
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.
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
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 1 − i in polar form.
Write the argument of −i.
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
The polar form of (i25)3 is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
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"))^2`
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 : i–888
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
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.
