Advertisements
Advertisements
प्रश्न
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
विकल्प
1
-1
2
-2
Advertisements
उत्तर
\[a - ib = \frac{3 - 4ix}{3 + 4ix}\]
\[ = \frac{3 - 4ix}{3 + 4ix} \times \frac{3 - 4ix}{3 - 4ix}\]
\[ = \frac{9 + 16 x^2 i^2 - 24xi}{9 - 16 x^2 i^2}\]
\[ = \frac{\left( 9 - 16 x^2 \right) - i\left( 24x \right)}{9 + 16 x^2}\]
\[ \Rightarrow \left| a - ib \right|^2 = \left| \frac{\left( 9 - 16 x^2 \right) - i\left( 24x \right)}{9 + 16 x^2} \right|^2 \]
\[ \Rightarrow a^2 + b^2 = \frac{\left( 9 - 16 x^2 \right)^2 + \left( 24x \right)^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = \frac{81 + 256 x^4 - 288 x^2 + 576 x^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = \frac{81 + 256 x^4 + 288 x^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = \frac{\left( 9 + 16 x^2 \right)^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = 1\]
Hence, the correct option is (a).
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
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:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write 1 − i in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Show that 1 + i10 + i20 + i30 is a real number
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
