Advertisements
Advertisements
प्रश्न
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
उत्तर
\[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2 = \left( a z_1 - b z_2 \right)\left( \bar{{a z_1 - b z_2}} \right) + \left( a z_2 + b z_1 \right)\left( \bar{{a z_2 + b z_1}} \right)\]
\[ = \left( a z_1 - b z_2 \right)\left( a \bar{{z_1}} - b \bar{{z_2}} \right) + \left( a z_2 + b z_1 \right)\left( a \bar{{z_2}} + b \bar{{z_1}} \right)\]
\[ = \left( a^2 z_1 \bar{{z_1}} - ab z_1 \bar{{z_2}} - ab z_2 \bar{{z_1}} + b^2 z_2 \bar{{z_2}} \right) + \left( a^2 z_2 \bar{{z_2}} + ab z_1 \bar{{z_2}} + ab z_2 \bar{{z_1}} + b^2 z_1 \bar{{z_1}} \right)\]
\[ = \left[ \left( a^2 + b^2 \right) z_1 \bar{{z_1}} + \left( a^2 + b^2 \right) z_2 \bar{{z_2}} \right]\]
\[ = \left[ \left( a^2 + b^2 \right)\left( z_1 \bar{{z_1}} + z_2 \bar{{z_2}} \right) \right]\]
\[ = \left[ \left( a^2 + b^2 \right)\left( \left| z_1 \right|^2 + \left| z_2 \right|^2 \right) \right]\]
Hence,
\[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2 = \left( a^2 + b^2 \right)\left( \left| z_1 \right|^2 + \left| z_2 \right|^2 \right)\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i30 + i80 + i120
Find the value of the following expression:
i5 + i10 + i15
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
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:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
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.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Write 1 − i in polar form.
Write the argument of −i.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
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 i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
Find a and b if (a + ib) (1 + i) = 2 + i
Evaluate the following : i35
Evaluate the following : i93
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
Show that `(-1+sqrt3i)^3` is a real number.
