Advertisements
Advertisements
Question
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Advertisements
Solution
Let \[z = x + iy\]
Then,
\[z + 1 = \left( x + 1 \right) + iy\]
\[ \Rightarrow \left| z + 1 \right| = \sqrt{\left( x + 1 \right)^2 + y^2}\]
\[\therefore \left| z + 1 \right| = z + 2\left( 1 + i \right)\]
\[ \Rightarrow \sqrt{x^2 + 2x + 1 + y^2} = \left( x + iy \right) + 2 + 2i\]
\[ \Rightarrow \sqrt{x^2 + 2x + 1 + y^2} = \left( x + 2 \right) + i\left( y + 2 \right)\]
\[ \Rightarrow \sqrt{x^2 + 2x + 1 + y^2} = \left( x + 2 \right) \text { and } y + 2 = 0\]
\[ \Rightarrow x^2 + 2x + 1 + y^2 = \left( x + 2 \right)^2 \text { and } y = - 2\]
\[ \Rightarrow x^2 + 2x + 1 + y^2 = x^2 + 4x + 4 \text { and } y = - 2\]
\[ \Rightarrow y^2 = 2x + 3 \text { and } y = - 2\]
\[ \Rightarrow 4 = 2x + 3 \text { and } y = - 2\]
\[ \Rightarrow 2x = 1 \text { and } y = - 2\]
\[ \Rightarrow x = \frac{1}{2} \text { and } y = - 2\]
\[\therefore z = x + iy = \frac{1}{2} - 2i\]
Thus,
\[z = \frac{1}{2} - 2i\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i9 + i19
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:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i30 + i80 + i120
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:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
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
\[(x + iy)(2 - 3i) = 4 + i\]
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 \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
Solve the equation \[\left| z \right| = z + 1 + 2i\].
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write −1 + i \[\sqrt{3}\] in polar form .
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
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 `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Evaluate the following : i888
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+ sqrt(3)i)^3` is a real number.
