An equation of the tangent plane to the surface z=f(x,y) at the point P(x0,y0,z0)
is:
Fx′(x0,y0,z0)(x−x0)+Fy′(x0,y0,z0)(y−y0)
+Fz′(x0,y0,z0)(z−z0)=0 where F(x,y,z)=0
Given
ax2+by2=2zF(x,y,z)=ax2+by2−2z=0 Then
2ax0(x−x0)+2by0(y−y0)−2(z−z0)=0
ax0x+by0y−z=ax02+by02−z0
ax0x+by0y−z=2z0−z0
If ux+vy++wz=p is a tangent plane to the paraboloid ax2+by2=2z, then
uax0=vby0=w−1=pz0
x0=−awu,y0=−bwv,z0=−wpPutting these values in equation of the paraboloid
a(−awu)2+b(−bwv)2=2(−wp)
(au2)+(bv2)+2pw=0
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