Complex NumbersMedium

Question

Let z be the complex number satisfying $|z - 5| \leq 3$ and having maximum positive principal argument.

Then $34\left| \frac{5z - 12}{5iz + 16} \right|^{2}$ is equal to:

Options

A.16
B.12
C.26
D.20

Solution

$$|z - 5| \leq 3 $$For $arg(z)$ to be maximum, z lies at P .

$${z \equiv (4cos\theta,4sin\theta) }{\equiv \left( 4 \cdot \left( \frac{4}{5} \right),4\left( \frac{3}{5} \right) \right) = \left( \frac{16}{5},\frac{12}{5} \right) = \frac{16}{5} + \frac{12i}{5} }$$Now, $34\left| \frac{5z - 12}{5iz + 16} \right|^{2} = 34\left| \frac{(16 + 12i) - 12}{(16i - 12) + 16} \right|^{2}$

$${= 34\left| \frac{4 + 12i}{16i + 4} \right|^{2} }{= 34\left( \frac{16 + 144}{256 + 16} \right) = 34\left( \frac{160}{272} \right) = 20}$$

Create a free account to view solution

View Solution Free
Topic: Complex Numbers·Practice all Complex Numbers questions

More Complex Numbers Questions

If z1 and z2 are two conjugate complex numbers and amp(z1) = θ , then amp (z1) + amp (z2) and amp (z1 / z2) are equ...If amp (z) = θ, then amp (1/z) is equal to-...If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{- 1}$ are two roots of the equation$x^{3} + {ax}^{2} + bx ...For all complex numbers z1, z2 satisfying |z1|=12 and |z2 - 3 - 4i | = 5, the minimum value of |z1 - z2 | is...If |z - z1| = |z1| and |z - z2| = |z2| be the of two circles if the two circles touch each other then...