Answer
No
Work Step by Step
Our aim is to determine the tangent plane equation.
The general form is: $(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$ ...(1)
From the given data, we have $f(x,y,z)=(1,1,-1)$
Equation (1), becomes:
Thus,
$(x-x_0)(2x)+(y-y_0)(-2y)+(z-z_0)(-2z)=0$
There must exist a constant number $c$ when the planes are parallel.
Therefore, $(\dfrac{c}{2})^2-(-\dfrac{c}{2})^2-(-\dfrac{c}{2})^2=-\dfrac{c^2}{4}$
This yields no solution.
Thus, No point can be found on the hyperboloid.
