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

Answer: 4

$C_1:4|z-2|=|z+\bar{z}-16|$
$16\left((x-2{)}^2+y^2\right)=(2x-16{)}^2$
$\Rightarrow 4\left(x^2-2x+1+y^2\right)=(x-8{)}^2$
$\Rightarrow x^2+4y^2=48$
$\Rightarrow \frac{x^2}{16}+\frac{y^2}{12}=1$
$C_2:\arg \left(\frac{z-1}{z+1}\right)=\pm \frac{\pi }{2}$
$\Rightarrow x^2+y^2=1$
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
$\therefore \text{ Number of tangents }=12\equiv m$
$\Rightarrow \frac{m}{3}=4$