Show that the angle between the tangent at any point P and the line joining P to the origin O is same at all points on the curve `log(x^2+y^2)=ktan^(-1)(y/x)`

Explanation

Let `P(x, y)` be a point on the curve


`log(x^{2}+y^{2})=k \tan ^{-1} \frac{y}{x}`.


Differentiating both sides with respect to `x`, we get
$$
\begin{aligned}
&\frac{2 x+2 y y^{\prime}}{\left(x^{2}+y^{2}\right)}=\frac{k\left(x y^{\prime}-y\right)}{\left(x^{2}+y^{2}\right)} \\
&\Rightarrow y^{\prime}=\frac{2 x+k y}{k x-2 y}=m_{1}(\text { say })
\end{aligned}
...