Answer to Question #202784 in Analytic Geometry for tanya

Question

Show that if ux+ vy+ wz = p is a tangent plane to the paraboloid ax² +by² = 2z, then

(u²/a) + (v²/b) + 2pw = 0.

Answer 1

An equation of the tangent plane to the surface $z=f(x, y)$ at the point $P(x_0, y_0, z_0)$

is:

$$F'_x(x_0, y_0, z_0)(x-x_0)+F'_y(x_0, y_0, z_0)(y-y_0)$$

$$+F'_z(x_0, y_0, z_0)(z-z_0)=0$$

where $F(x, y, z)=0$

Given

$$ax^2+by^2=2z$$ $$F(x, y, z)=ax^2+by^2-2z=0$$

Then

$$2ax_0(x-x_0)+2by_0(y-y_0)-2(z-z_0)=0$$

$$ax_0x+by_0y-z=ax_0^2+by_0^2-z_0$$

$$ax_0x+by_0y-z=2z_0-z_0$$

If $ux+vy++wz=p$  is a tangent plane to the paraboloid $ax^2+by^2=2z,$ then

$$\dfrac{ax_0}{u}=\dfrac{by_0}{v}=\dfrac{-1}{w}=\dfrac{z_0}{p}$$

$$x_0=-\dfrac{u}{aw}, y_0=-\dfrac{v}{bw}, z_0=-\dfrac{p}{w}$$

Putting these values in equation of the paraboloid

$$a(-\dfrac{u}{aw})^2+b(-\dfrac{v}{bw})^2=2(-\dfrac{p}{w})$$

$$(\dfrac{u^2}{a})+(\dfrac{v^2}{b})+2pw=0$$