Show that if ux+ vy+ wz = p is a tangent plane to the paraboloid ax2 +by2 = 2z, then
(u2/a) + (v2/b) + 2pw = 0.
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'_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
"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
"x_0=-\\dfrac{u}{aw}, y_0=-\\dfrac{v}{bw}, z_0=-\\dfrac{p}{w}"
Putting these values in equation of the paraboloid
"(\\dfrac{u^2}{a})+(\\dfrac{v^2}{b})+2pw=0"
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