Let z 1 and z 2 be two complex numbers such that | z 1 | = 1 and | z 2 | = 10 . If = arg ( z 1 - z 2 z 2 ) then maximum value of tan 2 is

Question

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $| z_{1} |$ $= 1$ and $| z_{2} |$ $= 10$ . If $θ = \arg (\frac{z_{1} - z_{2}}{z_{2}})$ then maximum value of $\tan^{2} θ$ is

Smart Achievers · Accepted Answer

(3)

$Put \frac{z_{1}}{z_{2}} = z$

$∴$ $| z_{2} |$ $θ = \arg (\frac{z_{1} - z_{2}}{z_{2}})$ $| z |$ $\Rightarrow$ $| z |$ $= \frac{1}{10}$

$Also, θ = \arg (\frac{z_{1}}{z_{2}} - 1) = \arg (z - 1)$

$∴$ $\tan^{2} θ_{\max} = \frac{1 / 100}{99 / 100} = \frac{1}{99}$

Solution

Q.
Let $z_{1}$ and $z_{2}$ be two complex numbers such that $| z_{1} |$ $= 1$ and $| z_{2} |$ $= 10$ . If $θ = \arg (\frac{z_{1} - z_{2}}{z_{2}})$ then maximum value of $\tan^{2} θ$ is

1 $\frac{1}{10}$

2 $\frac{1}{100}$

3 $\frac{1}{99}$

4 $\frac{10}{99}$

Ans.
(3)

$Put \frac{z_{1}}{z_{2}} = z$

$∴$ $| \frac{z_{1}}{z_{2}} |$ $=$ $| z |$ $\Rightarrow$ $| z |$ $= \frac{1}{10}$

$Also, θ = \arg (\frac{z_{1}}{z_{2}} - 1) = \arg (z - 1)$

$∴$ $\tan^{2} θ_{\max} = \frac{1 / 100}{99 / 100} = \frac{1}{99}$

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let z1 and z2 be two complex numbers such that |z1|=1 and |z2|=10. If θ=arg(z1-z2z2) then maximum value of tan2θ is

1 110

2 1100

3 199

4 1099

Ans. (3) Put z1z2=z ∴ |z1z2|=|z|⇒|z|=110 Also, θ=arg(z1z2-1)=arg(z-1) ∴ tan2θmax=1/10099/100=199