Working with circles? (ellipse) Implicit differentiation

[Higg152]

Hey everyone, 3 years out of the game and I feel super rusty : (

I have this equation of an ellipse (x^2)+xy+(y^2) = c

Where c is any positive real number.

Need to find the coordinates of all points on the curve with a horizontal tangent line.

If someone can show the steps & thought process to solving this, that would be great!
Thanks!

 

[Deleted member 4993]

Hey everyone, 3 years out of the game and I feel super rusty : (

I have this equation of an ellipse (x^2)+xy+(y^2) = c

Where c is any positive real number.

Need to find the coordinates of all points on the curve with a horizontal tangent line.

If someone can show the steps & thought process to solving this, that would be great!
Thanks!

(x^2)+xy+(y^2) = c

By implicit differentiation - as implied by the title..

2*x + y + x*y' + 2*y*y' = 0

solve for y' and set y' = o

and continue.....

 

[Bob Brown MSEE]

(x - a) (x - a) + (x - a) (y - b) + (y - b) (y - b) = a^2+a b+b^2-2 a x-b x+x^2-a y-2 b y+x y+y^2

Noting the coefficients of x we get b=-2a (when pt is moved to origin and y=0x is linear term).
So solutions lie on the line y=-2x

Try xx+x(-2x)+(-2x)(-2x) = n
to get both x then y=-2x

 

[Higg152]

(x - a) (x - a) + (x - a) (y - b) + (y - b) (y - b) = a^2+a b+b^2-2 a x-b x+x^2-a y-2 b y+x y+y^2

Noting the coefficients of x we get b=-2a (when pt is moved to origin and y=0x is linear term).
So solutions lie on the line y=-2x

Try xx+x(-2x)+(-2x)(-2x) = n
to get both x then y=-2x

Hey Bob, my answer came out as y=2x. Where do you think the difference came about?

The final part of the answer involves +-sqrt(c/3), which means we can't see whether it is y=2x or y=-2x through the final answer.

 

[stapel]

Hey Bob, my answer came out as y=2x. Where do you think the difference came about?

Until you show your work, we'll have no way even of guessing where the differences may have arisen. Sorry.

Please reply with the missing information. Thank you!