Question

Consider a conic on the complex plane represented by the equation $|z-3-4i|=\frac{1}{\sqrt{2}}\left|\frac{(1-i)z+(1+i)\overline{z}+2}{2}\right|$ 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

Answer: 8

$|z-3-4i|=\frac{1}{\sqrt{2}}\left|\frac{(1-i)z+(1+i)\overline{z}+2}{2}\right|$
put $z=x+iy$
$\sqrt{(x-3{)}^2+(y-4{)}^2}=\frac{1}{2}\left|\frac{(1-i)(x+iy)+(1+i)(x-iy)+2}{2}\right|$
$\sqrt{(x-3{)}^2+(y-4{)}^2}=\left|\frac{x+y+1}{\sqrt{2}}\right|\text{ (Equation of parabola) }$
$SP=ePM$
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)
$=8\sqrt{2}\equiv \lambda \sqrt{2}\Rightarrow \lambda =8$