Finding tanget line equal to zero and parallel to y=x

[phawksbball24]

I am stuck and any help would be greatly appreciated.

The equation x^2+xy+y^2=1 is given.

First you are suposed to find dy/dx

which I did and got dy/dx= (-2x-y) / (x+2y)

Then you are suposed to find the points where the tangent line is horizontal so I know that dy/dx must equal 0 and this is where I'm not sure if I am doing it correctly.

I made -2x-y=0 and solved that to get y=-2x (Now I don't know what to do in

order to figure out what those points actually are on the elipse)

Same thing for when I try to get the tangent line to be parallel to y=x. I know the slope must be 1 so I made

-2x-y=x+2y and then solved that to get y=-x (however once again I am stuck

because I am not sure what to do to find the values of the points on the ellipse)

I hope I am on the right track and any help you could give me to help me finish the problem would be greatly appreciated, thanks.

 

[o_O]

You've done most of the work already! Good job!

For the horizontal tangent line, you got y = -2x. Now plug this back into your original equation to find the point(s) where the tangent line touches:

x² +x(-2x) + (-2x)² = 1

And you do the same thing for the parallel tangent line.

 

[phawksbball24]

Alright awesome, thanks a ton!