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)z=f(x, y) at the point P(x0,y0,z0)P(x_0, y_0, z_0)

is:



Fx(x0,y0,z0)(xx0)+Fy(x0,y0,z0)(yy0)F'_x(x_0, y_0, z_0)(x-x_0)+F'_y(x_0, y_0, z_0)(y-y_0)

+Fz(x0,y0,z0)(zz0)=0+F'_z(x_0, y_0, z_0)(z-z_0)=0

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

Given

ax2+by2=2zax^2+by^2=2zF(x,y,z)=ax2+by22z=0F(x, y, z)=ax^2+by^2-2z=0

Then


2ax0(xx0)+2by0(yy0)2(zz0)=02ax_0(x-x_0)+2by_0(y-y_0)-2(z-z_0)=0

ax0x+by0yz=ax02+by02z0ax_0x+by_0y-z=ax_0^2+by_0^2-z_0

ax0x+by0yz=2z0z0ax_0x+by_0y-z=2z_0-z_0

If ux+vy++wz=pux+vy++wz=p  is a tangent plane to the paraboloid ax2+by2=2z,ax^2+by^2=2z, then


ax0u=by0v=1w=z0p\dfrac{ax_0}{u}=\dfrac{by_0}{v}=\dfrac{-1}{w}=\dfrac{z_0}{p}

x0=uaw,y0=vbw,z0=pwx_0=-\dfrac{u}{aw}, y_0=-\dfrac{v}{bw}, z_0=-\dfrac{p}{w}

Putting these values in equation of the paraboloid


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

(u2a)+(v2b)+2pw=0(\dfrac{u^2}{a})+(\dfrac{v^2}{b})+2pw=0


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