Find an equation of the plane tangent to the following...

Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations).
$$z^{2}-x^{2} / 16-y^{2} / 9-1=0 ;(4,3,-\sqrt{3}) \text { and }(-8,9, \sqrt{14})$$

Key Concepts

Gradient Vector

The gradient vector of a multivariable function is composed of its partial derivatives and points in the direction of the greatest rate of change of the function. In the context of tangent planes for implicit surfaces, the gradient vector at a point is perpendicular to the surface, providing the normal vector needed to determine the plane.

Tangent Plane

The tangent plane to a surface at a given point is the plane that best approximates the surface locally. It can be computed using the point of tangency and the normal vector (obtained from the gradient of the defining function), representing the linearization of the surface.

Point-Normal Form of a Plane

The point-normal form is a method for writing the equation of a plane by specifying a point on the plane and a normal vector to it. This form is particularly useful in multivariable calculus when deriving the equation of a tangent plane from the normal determined by the gradient.

Implicit Differentiation on Surfaces

Implicit differentiation is used to derive relationships between variables in an equation defining a surface implicitly. By differentiating the defining equation, one can obtain the gradient vector which is essential for finding the normal vector and hence the tangent plane.

Level Surfaces

Level surfaces are sets of points where a multivariable function has a constant value. The gradient vector of the function is always normal to these level surfaces, making this concept critical in understanding how to derive equations for tangent planes on such surfaces.

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