Question

The points in the $set\{zϵC:arg\left(\frac{z-2}{z-6i}\right)=\frac{\pi }{2}\}$ lie on the curve which is a (where C denotes the set of all complex numbers) :

• A. Circle
• B. Pair of Straight lines
• C. Parabola
• D. Hyperbola

Answer 1

Answer: (A)

Circle

To find the Points in the set lie on the curve
$Arg\left(\frac{z-2}{z-6i}\right)=\frac{\pi }{2}$
$\left(\frac{z-2}{z-6i}\right)=tan\frac{\pi }{2}$
$\left(\frac{x+iy-2}{x+iy-6i}\right)=tan\left(\frac{\pi }{2}\right)$

$\left(\frac{x-2+iy}{x+i(y-6)}\right)=undefined$

$\left(\frac{x-2+iy}{x+i(y-6)}\right)\times \left(\frac{x-i(y-6)}{x-i(y-6)}\right)=\frac{FiniteNumber}{0}$

$\left(\frac{(x-2+iy)(x-i(y-6\left)\right)}{x^2+(y-6{)}^2}\right)=\frac{FiniteNumber}{0}$

From this we get, $x^{{}^2}+(y-6{)}^2=0$ which is a equation of Circle.

Hence , Option A