Advertisements
Advertisements
प्रश्न
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|.
उत्तर
Distance of the point representing z from the centre of the circle is |z – (–4 + i0)| = |z + 4|.
According to given condition |z + 4| ≤ 3.
Now |z + 1| = |z + 4 – 3| ≤ |z + 4| + |–3| ≤ 3 + 3 = 6
Therefore, greatest value of |z + 1| is 6.
Since least value of the modulus of a complex number is zero, the least value of |z + 1| = 0.
APPEARS IN
संबंधित प्रश्न
Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.
Find the modulus of `(1+i)/(1-i) - (1-i)/(1+i)`
Find the conjugate of the following complex number:
\[\frac{(3 - i )^2}{2 + i}\]
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:
\[\frac{1 - i}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 + 2i}{1 - 3i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{- 16}{1 + i\sqrt{3}}\]
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].
Write the conjugate of \[\frac{2 - i}{\left( 1 - 2i \right)^2}\] .
If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=
If \[\frac{( a^2 + 1 )^2}{2a - i} = x + iy, \text { then } x^2 + y^2\] is equal to
If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x2 + y2 =
If \[\frac{1 - ix}{1 + ix} = a + ib\] then \[a^2 + b^2\]
If |z2 – 1| = |z|2 + 1, then show that z lies on imaginary axis.
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 ______.
What is the conjugate of `(sqrt(5 + 12i) + sqrt(5 - 12i))/(sqrt(5 + 12i) - sqrt(5 - 12i))`?
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find arg`(z_1/z_4)` + arg`(z_2/z_3)`.
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?
