Question

Find the equation of the tangent plane to the hyperboloid f(x,y,z)=x^2/a^2 +y^2/b^2 -z^2/c^2 =1
at (x_0,y_0,z_0 )

Answer 1

Take the gradient which gives you a vector perpendicular to the surface. Evaluate it at (x_0,y_0,z_0) and you'll have a normal vector. From there you need to evaluate:
\[\hat{n} \cdot (\vec{r} - (x_0,y_0,z_0))=0\]
And this will give you the equation of the plane.

Answer 2

The gradient is defined as:
\[\vec{\nabla}f(x_1,…,x_n)=\sum_{k=1}^n \frac{\partial f}{\partial x_k}\vec{e}_k\]
where e_k is the unit vector in the kth direction.

Answer 3

i have already calculate it..can you please check it.