Question

Implicit Differentiation:
Need help simplifying...

Answer 1

\[x^3+xy^2+ay^2-3ax^2=0\]
This is a curve and note that a is a positive constant. Find the maximum point on the curve is called M. Find the x-coordinate of M in terms of a

I have already successfully differentiated and equated dy/dx to 0
\[(2xy+2ay)\frac{ dy }{ dx } + 3x^2 + y^2-6ax=0\]
So we get
\[3x^2+y^2-6ax=0\]
I'm stuck on this part on how to eliminate y^2 from the equation so I can express x in terms of a...

Please help!

Answer 2

If we look back at the original equation:\[\Large\rm x^3+y^2(x+a)-3ax^2=0\]Solve for y^2 here,
That should get the job done, yes? :o

Answer 3

So we solve for y^2 from the original equation and then plug it into the final equation?

Answer 4

Yah, that seems right :)

Answer 5

So I've got
\[y^2=\frac{ 3ax^2-x^3 }{ x+a }\]
Which I then plugged in. Then I multiplied the equation by (x+a) to remove the fraction to get:
\[2x^3+6a^2x = 0\]
I'm not to sure how to express x in terms of a from here...

Answer 6

Ok I just tried this:
\[2x^3=-6a^2x\]
\[2x^2=-6a^2\]
\[x^2=-3a^2\]
\[x=\sqrt{-3a^2}\]
\[x=\sqrt{-3}a\]
Is the negative 3 correct? I don't think this question involves complex numbers...

Answer 7

Oh I ended up with: \(\Large\rm 2x^3-6a^2x=0\)
Not plus in the middle.
Hmm...

Answer 8

I think you're correct just checked my working and forgot to multiply by the negative when expanding...
So the answer is
x= sqrt3 a
Which I believe is correct! Thanks so much for your help!!! So much algebra for 2 marks -_-

Answer 9

lol ikr D: easy differentiation, rest of that was a pain though!

Answer 10

Yeap lol!