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

if ux+vy+wz=p...................(1)

is a tangent plane to the paraboloid

ax²+by²=2z................(2)

and let (l,m,n) be the point of contact of tangent plane and paraboloid.

So as per condition of tangency of a plane to a conicoid ,

we get,

alx+bmy=z+n

or

alx+bmy-z=n.....(3)

This is a tangent plane to the given paraboloid and hence it is same as plane given in equation(2).

So,

by equation (1) and (3),

we get,

$\frac{al}{u}=\frac{bm}{v}=\frac{-1}{w}=\frac{n}{p}$

or l=-u/wa

m=-v/wb

n=-p/w.......(4)

these values of (l,m,n) are point of contact and satisfy the equation of paraboloid.

Therefore,putting these values in equation (2)

we get,

a(-$\frac{u}{aw})$²+b(-$\frac{v}{wb})$² =2(-$\frac{p}{w})$

on solving this,

we get

 u²/a+v²/b+2pw=0

which is the required answer.