The tangent plane to the surface $z = x \sin ( x / y ) , y \neq 0$ at any point passes through the origin.

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The tangent plane to the surface at any point has the equation $z = f(x,y) + f_x(x,y)(x-x_0) + f_y(x,y)(y-y_0)$, where $f(x,y)$ is the surface equation and $f_x(x,y)$ and $f_y(x,y)$ are the partial derivatives of $f(x,y)$. In this case, $f(x,y) = x \sin ( x / y )$, so $f_x(x,y) = \sin ( x / y ) + x \cos ( x / y ) / y$ and $f_y(x,y) = -x \sin ( x / y ) / y^2$. At any point $(x_0,y_0)$, the tangent plane equation becomes $$z = x_0 \sin ( x_0 / y_0 ) + ( \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 ) / y_0 )(x-x_0) - (x_0 \sin ( x_0 / y_0 ) / y_0^2 )(y-y_0)$$$$z = x_0 \sin ( x_0 / y_0 ) + \sin ( x_0 / y_0 )(x-x_0) + x_0 \cos ( x_0 / y_0 ) (x-x_0)/y_0 - x_0 \sin ( x_0 / y_0 ) (y-y_0)/y_0^2$$$$z = x_0 \sin ( x_0 / y_0 ) + \sin ( x_0 / y_0 )x - \sin ( x_0 / y_0 )x_0 + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )y_0/y_0^2$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^2 + x_0 \sin ( x_0 / y_0 )$$$$z = \sin ( x_0 / y_0 )x - x_0 \sin ( x_0 / y_0 ) + x_0 \cos ( x_0 / y_0 )x/y_0 - x_0 \cos ( x_0 / y_0 )x_0/y_0 - x_0 \sin ( x_0 / y_0 )y/y_0^

The Tangent Plane to the Surface z=xsin⁡(x/y),y≠0z = x \sin ( x / y ) , y \neq 0z=xsin(x/y),y=0

The Tangent Plane to the Surface $z = x \sin ( x / y ) , y \neq 0$