Advertisements
Advertisements
प्रश्न
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
उत्तर
Let \[z = x + iy\]
Then ,
\[z^2 = \left( x + iy \right)^2 \]
\[ = x^2 + i^2 y^2 + 2ixy\]
\[ = x^2 - y^2 + 2ixy [ \because i^2 = - 1]\]
and
\[\left| z \right| = \sqrt{x^2 + y^2}\]
According to the question,
\[Re\left( z^2 \right) = 0 \text { and } \left| z \right| = 2\]
\[ \Rightarrow x^2 - y^2 = 0 \text { and } \sqrt{x^2 + y^2} = 2\]
\[ \Rightarrow x^2 - y^2 = 0 \text { and } x^2 + y^2 = 4\]
\[\text { On Adding both the equations, we get }\]
\[2 x^2 = 4\]
\[ \Rightarrow x^2 = 2\]
\[ \Rightarrow x = \pm \sqrt{2}\]
\[ \Rightarrow y^2 = 2\]
\[ \Rightarrow y = \pm \sqrt{2}\]
Thus,
\[x = \pm \sqrt{2} \text { and } y = \pm \sqrt{2}\]
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: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
(ii) i528
Evaluate the following:
\[\frac{1}{i^{58}}\]
Find the value of the following expression:
i + i2 + i3 + i4
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
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 + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
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.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
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 z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
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.
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
If z is a complex number, then
Which of the following is correct for any two complex numbers z1 and z2?
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if a + 2b + 2ai = 4 + 6i
Find a and b if abi = 3a − b + 12i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Evaluate the following : i–888
Show that `(-1 + sqrt3 "i")^3` is a real number.
