Let $f(t)$ be a differntiable function $\forall t $ , let $z$ be a surface such that $z=xf(\frac{x}{y}),y\neq 0$.
Find the common point of the all the tangent planes.
My attempt:
Denote $g(x,y):=xf(\frac{x}{y}),y\neq 0$.
I can find the tangent surface in $(x_0,y_0)$ using theorem $z=g(x_0,y_0)+\frac{dg}{dx}(x_0,y_0)(x-x_0)+\frac{dg}{dy}(x_0,y_0)(y-y_0).$
Now, $(x_0,y_0)=(t,y_0)$
$z=tf(\frac{t}{y_0})+(f(\frac{t}{y_0})+tf(\frac{t}{y_0})\frac{1}{y_0})(x-t)+tf(-\frac{t}{y_0^2})(y-y_0)$
How to find the common point of all tangent planes ?