Question

Consider two curves, $C_1: 4|z-2|=|z+\bar{z}-16|$ and $C_2: \arg \left(\frac{z-1}{z+1}\right)= \pm \frac{\pi}{2}$. Let tangent drawn to curve $C_2$ which meets the curve $C_1$ at $P\left(x_0, y_0\right)$ such that $x_0, y_0 \in I$. If number of such tangents is equal to $m$, then find the value of $\left(\frac{ m }{3}\right)$.

Answer 1

$C_1: 4|z-2|=|z+\bar{z}-16|$
$ 16\left(( x -2)^2+ y ^2\right)=(2 x -16)^2 $
$\Rightarrow 4\left( x ^2-2 x +1+ y ^2\right)=( x -8)^2 $
$\Rightarrow x ^2+4 y ^2=48$
$\Rightarrow \frac{ x ^2}{16}+\frac{ y ^2}{12}=1$
$C _2 : \arg \left(\frac{ z -1}{ z +1}\right)= \pm \frac{\pi}{2} $
$\Rightarrow x ^2+ y ^2=1$
Number of points on the ellipse with integral co-ordinates are 6 i.e. $( \pm 4,0),( \pm 2, \pm 3)$.
From each point, two tangents are drawn to the circle
$\therefore \text { Number of tangents }=12 \equiv m $
$\Rightarrow \frac{ m }{3}=4 $