Question

Let, $|z-z_1|=\left|z_1\right|and|z-z_2|=\left|z_2\right|$ be the equation of two circles. If the circles touch each other, then

• A. $Re\left(z_1{z}_2\right)=0$
• B. $Re\left(\frac{z_1}{z_2}\right)=0$
• C. $Img\left(z_1{z}_2\right)=0$
• D. $Img\left(\frac{z_1}{z_2}\right)=0$

Answer 1

The correct option is D $Img\left(\frac{z_1}{z_2}\right)=0$

$|z-z_1|=\left|z_1\right|$ $;$ $|z-z_2|=\left|z_2\right|$

$\Rightarrow \sqrt{z_2-z_1^2}=\sqrt{z_1^2}\Rightarrow z=\sqrt{2}{z}_1$

$\sqrt{z_2-z_2^2}=\sqrt{z_2^2}\Rightarrow z=\sqrt{2}{z}_2$

$\therefore z_1=z_2$ i.e, $z_1=z_2=0$

$(x_1-x_2)+i(y_1-y_2)=0$

$\Rightarrow Img\left(\frac{z_1}{z_2}\right)=0$

Hence, the answer is $Img\left(\frac{z_1}{z_2}\right)=0.$