The equation of the ellipsoid is \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1\).
Let the surface function be, \(F(x,y,z) = \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} - 1\). (1)
The equation of the tangent plane to the given surface at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) is defined by, \({F_x}\left( {{x_0},{y_0},{z_0}} \right)\left( {x - {x_0}} \right) + {F_y}\left( {{x_0},{y_0},{z_0}} \right)\left( {y - {y_0}} \right) + {F_z}\left( {{x_0},{y_0},{z_0}} \right)\left( {z - {z_0}} \right) = 0\). (2)
Take partial derivative with respect to \(x\) at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) in the equation (1),
\(\begin{aligned}{l}{F_x}(x,y,z) = \frac{\partial }{{\partial x}}\left( {\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} - 1} \right)\\ = \frac{1}{{{a^2}}}(2x)\\ = \frac{{2x}}{{{a^2}}}\end{aligned}\)
Thus, the value of \({F_x}\left( {{x_0},{y_0},{z_0}} \right) = \frac{{2{x_0}}}{{{a^2}}}\).
Take partial derivative with respect to \(y\) at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) in the equation (1),
\({F_y}(x,y,z) = \frac{\partial }{{\partial x}}\left( {\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} - 1} \right)\)
\(\begin{aligned}{l} = \frac{1}{{{b^2}}}(2y)\\ = \frac{{2y}}{{{b^2}}}\end{aligned}\)
Thus, the value of \({F_y}\left( {{x_0},{y_0},{z_0}} \right) = \frac{{2{y_0}}}{{{b^2}}}\).
Take partial derivative with respect to \(z\) at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) in the equation (1),
\({F_z}(x,y,z) = \frac{\partial }{{\partial x}}\left( {\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} - 1} \right)\)
\(\begin{aligned}{l} = \frac{1}{{{c^2}}}(2z)\\ = \frac{{2z}}{{{c^2}}}\end{aligned}\)
Thus, the value of \({F_z}\left( {{x_0},{y_0},{z_0}} \right) = \frac{{2{z_0}}}{{{c^2}}}\).
Substitute the respective values in the equation (2) and obtain,
Thus, it is verified that the equation of the tangent plane to the ellipsoid \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1\) at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) is \(\frac{{x{x_0}}}{{{a^2}}} + \frac{{y{y_0}}}{{{b^2}}} + \frac{{z{z_0}}}{{{c^2}}} = 1\).
