Partial derivatives boundery point problems

[Liquidxlax]

Homework Statement

find the largest distance and shortest distance from the origin to the conic whose equation is

6x² + 4xy +3y² - 28=0

and hence determine the lengths of the semi axes of this conic

Homework Equations

Lagrange identity

F= f + λφ = 0

distance = d² =x²+ y²+ z²

The Attempt at a Solution

This has been my problems since grade 11... i have a hard time isolating a single term.

Either i want to sub in an equation into the distance formula which is only 2d for this problem or i need to use the Lagrange identity.

if Lagrange

partial x = 2x + λ(12x +4y) = 0

partial y = 2y +λ(6y+4x) = 0

any help is appreciated.

 

[tiny-tim]

Hi Liquidxlax!

Wouldn't it be easier to just use polar coordinates?

 

[Dick]

I'm not sure how polar coordinates will help. Just go ahead an do it. Solve the first equation for x and substitute that into the second equation. What do you get? Does that tell you something about lambda?

 

[Liquidxlax]

I'm not sure how polar coordinates will help. Just go ahead an do it. Solve the first equation for x and substitute that into the second equation. What do you get? Does that tell you something about lambda?

what about the d^2? It has no known value, so I'm stuck still

 

[Dick]

what about the d^2? It has no known value, so I'm stuck still

Start with the linear equations you got from differentiating. They are the easier ones. Save the quadratic for last.

 

[Liquidxlax]

thanks for the help i did manage to solve it :D