Show that the surfaces $ \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2}$ and $ x^2 + y^2+ \left(z – \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + c^2)$ are tangent at the point $(0, ±b,c)$
To show that 2 surfaces are tangent, is it necessary and suficient to show that both points are in both surfaces and the tangent plane at those points is the same?
Because if we just show that both points are in both surfaces, the surfaces could just intercept each other. And if we just show the tangent plane is the same at 2 points of the surfaces, those points do not need to be the same.
Thanks!
Best Answer
The respective gradients of the surfaces are locally perpendicular to them: $$ \nabla f_1 = 2\left(\frac{x}{a^2}, \frac{y}{b^2}, -\frac{z}{c^2} \right) \\ \nabla f_2 = 2\left(x, y, z - \frac{b^2+c^2}{c}\right) $$ At any point of common tangency, the gradients are proportional. Therefore $\nabla f_1 = \lambda \nabla f_2$. This is seen to be true is you substitute $\left(0, \pm b, c\right)$ into the expressions for the gradients, the proportionality constant being $b^2$. QED.