Question

Consider a conic on the complex plane represented by the equation $| z -3-4 i |=\frac{1}{\sqrt{2}}\left|\frac{(1- i ) z +(1+ i ) \overline{ z }+2}{2}\right|$ then if the minimum length of the focal chord of the conic is $\lambda \sqrt{2}$ (where $\lambda \in N$ ) then find the value of $\lambda$.

Answer 1

$| z -3-4 i |=\frac{1}{\sqrt{2}}\left|\frac{(1- i ) z +(1+ i ) \overline{ z }+2}{2}\right|$
put $z = x + iy$
$\sqrt{(x-3)^2+(y-4)^2}=\frac{1}{2}\left|\frac{(1-i)(x+i y)+(1+i)(x-i y)+2}{2}\right| $
$\sqrt{(x-3)^2+(y-4)^2}=\left|\frac{x+y+1}{\sqrt{2}}\right| \text { (Equation of parabola) } $
$S P=e P M$
Hence, Focus $S \equiv(3,4)$
Directrix : $x+y+1=0$
Length of minimum focal chord = length of latus rectum
L.R $=2(\perp$ distance from focus on the directrix)
$=8 \sqrt{2} \equiv \lambda \sqrt{2} \Rightarrow \lambda=8 $