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A curve has equation $x^2 - 4xy + 2y^2 = 1$.

a) Find ad simplify an expression for $\frac{dy}{dx}$.

b) Show that the tangent to the curve at the point $P=(1,2)$ has the equation

$$3x - 2y +1 = 0.$$

The tangent to the curve at the point $Q$ is parallel to the tangent at $P$.

c) Find the coordinates of $Q$.

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Vales
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    Hi! If you're new to Math SE, welcome. Please tell us a little bit about how you came to this question. Also, by site policy you'll get much more comprehensive replies if you tell us what you attempted in solving the problem, and where you got stuck. Finally, with regards to your question, have you heard of the implicit function theorem? – Jonathan Y. Sep 29 '13 at 21:45
  • It is entirely likely he has not heard of that theorem as it is the unfortunate custom to assign such problems in first semester calculus without much mention of the theorem which justifies the calculation. Such is the state of calculus in America. – James S. Cook Sep 29 '13 at 21:49
  • @JamesS.Cook, would you then present the chain-rule and let freshmen solve simple cases by hand? Vales, that's part of the reason we need you to specify what you know about the topic at hand and what you've tried: so we can tailor the tools and terminology we apply to your framework. – Jonathan Y. Sep 29 '13 at 21:53
  • @JonathanY. of course, I would tell them that we assume it is possible to solve for $y$ as a function of $x$ locally, but in practice we cannot practically do such. Moreover, a theorem of advanced calculus justifies the calculation, but they'll have to wait for the proof. I don't think everybody says that much... in any event, I completely agree with your line of questioning. – James S. Cook Sep 29 '13 at 22:02
  • @JonathanY. if you have a minute or three, could you see what you think of http://math.stackexchange.com/questions/509510/c-h-edwards-advanced-calculus-of-several-variables-problem-3-5-of-page-194 Thanks! – James S. Cook Sep 30 '13 at 01:58

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This is going by @JamesS.Cook's remarks above; If you happen to be familiar with the implicit function theorem, we could be a little more exact.

Hint: for simplicity's sake, suppose we know that we can represent $y=y(x)$ such that $(x,y(x))$ is always a (unique, in the appropriate sense) point on that curve (in fact, such a representation is only possible in a small neighborhood of each point on that curve, but that needn't hinder us here). If we then define $$z(x) = x^2 -4xy(x) + 2(y(x))^2,$$ what does that say about $\frac{dz}{dx}$, and what would the chain rule then imply regarding $\frac{dy}{dx}$?

Jonathan Y.
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