Answer to Question #202784 in Analytic Geometry for tanya

Question #202784

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

(u2/a) + (v2/b) + 2pw = 0.


1
Expert's answer
2021-06-08T13:11:19-0400

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"


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