Question

If the points represented by complex numbers $z_1=a+ib,z_2=c+id$ and $z_1-z_2$ are collinear, then

• A. $ad+bc=0$
• B. $ad-bc=0$
• C. $ab+cd=0$
• D. $ab-cd=0$

Answer 1

Answer: (B)

$ad-bc=0$

Since
$z_1,z_2$ and $z_1-z_2$ are collinear.
$|z_1-z_2|=|z_1|-|z_2|$
$|(a-c)+i(b-d)|=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}$
$(a-c{)}^2+(b-d{)}^2=a^2+b^2+c^2+d^2+2\sqrt{(a^2+b^2)(c^2+d^2)}$
$a^2+c^2+b^2+d^2-2(ac+bd)=a^2+b^2+c^2+d^2+2\sqrt{(a^2+b^2)(c^2+d^2)}$
Hence
$-2(ac+bd)=2\sqrt{(a^2+b^2)(c^2+d^2)}$
$a^2{c}^2+b^2{d}^2+2abcd=a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2$
$a^2{d}^2+b^2{c}^2-2abcd=0$
$(ad-bc{)}^2=0$
$ad-bc=0$