Advertisements
Advertisements
Question
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Options
x−axis
circle with centre (−1, 0) and radius 1
y−axis
none of these
Advertisements
Solution
\[\left| z + 1 \right| = 1\]
\[ \Rightarrow \left| z + 1 \right|^2 = 1^2 \]
\[ \Rightarrow \left( z + 1 \right) \bar{\left( z + 1 \right)} = 1\]
\[ \Rightarrow \left( z + 1 \right)\left( \bar{z} + 1 \right) = 1\]
\[ \Rightarrow z \bar{z} + z + \bar{z} + 1 = 1\]
\[ \Rightarrow z \bar{z} + z + \bar{z} = 0\]
\[\text { Since }, z = x + iy\]
\[ \therefore z \bar{z} + z + \bar{z} = 0\]
\[ \Rightarrow \left( x + iy \right)\left( x - iy \right) + x + iy + x - iy = 0\]
\[ \Rightarrow x^2 + y^2 + 2x = 0\]
\[ \Rightarrow \left( x + 1 \right)^2 + \left( y - 0 \right)^2 = 1^2 \]
\[\text { which is the equation of a circle with centre } ( - 1, 0) \text { and radius }1\]
Hence, the correct option is (b).
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Evaluate the following:
i457
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the value of the following expression:
i30 + i80 + i120
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:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
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.
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Write (i25)3 in polar form.
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 \[z_1 = - \bar{{z_2}}\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the argument of −i.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The value of \[(1 + i )^4 + (1 - i )^4\] is
Find a and b if a + 2b + 2ai = 4 + 6i
Find a and b if (a – b) + (a + b)i = a + 5i
Evaluate the following : i93
Evaluate the following : i30 + i40 + i50 + i60
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
