Answer to Question #91558 in Analytic Geometry for Ra

Question #91558
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
2019-07-19T11:10:57-0400

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.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS