Advertisements
Advertisements
प्रश्न
Solve the equation `z^2 = barz`, where z = x + iy.
उत्तर
`z^2 = barz`
⇒ x2 – y2 + i2xy = x – iy
Therefore, x2 – y2 = x ......(1)
And 2xy = –y ......(2)
From (2), we have y = 0 or x = `- 1/2`
When y = 0, from (1)
We get x2 – x = 0
i.e., x = 0 or x = 1.
When x = `-1/2`, from (1)
We get y2 = `1/4 + 1/2` or y2 = `3/4`.
i.e., y2 = `+- sqrt(3)/2`
Hence, the solutions of the given equation are 0 + i0, 1 + i0, `-1/2 +i sqrt(3)/2`, `-1/2 -i sqrt(3)/2`
APPEARS IN
संबंधित प्रश्न
Find the conjugate of the following complex number:
\[\frac{1}{3 + 5i}\]
Find the conjugate of the following complex number:
\[\frac{(3 - i )^2}{2 + i}\]
Find the conjugate of the following complex number:
\[\frac{(1 + i)(2 + i)}{3 + i}\]
Find the modulus of \[\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}\].
Find the modulus and argument of the following complex number and hence express in the polar form:
1 + i
Find the modulus and argument of the following complex number and hence express in the polar form:
1 − i
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 - i}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1}{1 + i}\]
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].
If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=
If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x2 + y2 =
If |z2 – 1| = |z|2 + 1, then show that z lies on imaginary axis.
If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (–4, 0), find the greatest and least values of |z + 1|.
The conjugate of the complex number `(1 - i)/(1 + i)` is ______.
If a complex number lies in the third quadrant, then its conjugate lies in the ______.
If z1 = `sqrt(3) + i sqrt(3)` and z2 = `sqrt(3) + i`, then find the quadrant in which `(z_1/z_2)` lies.
What is the conjugate of `(sqrt(5 + 12i) + sqrt(5 - 12i))/(sqrt(5 + 12i) - sqrt(5 - 12i))`?
Solve the system of equations Re(z2) = 0, z = 2.
State True or False for the following:
If z is a complex number such that z ≠ 0 and Re(z) = 0, then Im(z2) = 0.
What is the conjugate of `(2 - i)/(1 - 2i)^2`?
If `(a^2 + 1)^2/(2a - i)` = x + iy, what is the value of x2 + y2?
If z = x + iy lies in the third quadrant, then `barz/z` also lies in the third quadrant if ______.
