Finding an Equation of a Tangent Plane to a Surface

Find an equation of the tangent plane to the hyperboloid of one sheet \( x^{2}+y^{2}-z^{2}=24\) at the point \((3,-4,1)\).

Solution The surface is given by the function \( F(x,y,z)=x^{2}+y^{2}-z^{2}-24=0.\) The partial derivatives of \(F\) are \[ F_{x}(x,y,z)=2x\qquad F_{y}(x,y,z)=2y\qquad F_{z}(x,y,z)=-2z \]

At the point \((3,-4,1)\), the partial derivatives are \[ F_{x}(3,-4,1)=6\qquad F_{y}(3,-4,1)=-8\qquad F_{z}(3,-4,1)=-2 \]

An equation of the tangent plane at \((3,-4,1)\) is \[ \begin{array}{rcl@{\quad}l} 6(x-3)-8(y+4)-2(z-1)& =& 0 & \\[4pt] 3x-4y-z& =&24 & \end{array} \]