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Find the coordinates of the points where the line tangent to the curve $$x^2-2xy+2y^2=4$$ is parallel to the $x$-axis, given that $$\frac{dy}{dx}=\frac{y-x}{2y-x}$$

By letting $dy/dx = 0$ I get $y=x$ which is no help... what do I do?

Thanks

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  • $\begingroup$ Plug $y=x$ into the original equation and solve for x and y. $\endgroup$ – Christopher Marley May 30 '18 at 4:41
  • $\begingroup$ Welcome to Mathematics Stack Exchange! I really like the lesson that this question teaches, very interesting $\ddot\smile$ $\endgroup$ – gen-z ready to perish May 30 '18 at 5:11
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All points on the curve obey the equation $x^2 - 2xy + 2y^2 = 4$. Having $y=x$ when $\frac{dy}{dx} = 0$ reduces this to $x^2 - 2x^2 + 2x^2 = 4$ or $x^2= 4$ or $x=2$ or $x=-2$, which you can find the possible $y$-coordinates for by using the equation again (substitute $x=2$ and solve for $y$, then the same for $x=-2$) so you do get your points that way. Don't forget the base equation after you compute differentials!

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  • $\begingroup$ My problem was that I didn’t think I was able to substitute x=y into the original equation and then substitute the x value back into the same equation. Logically if x =y and x=2,-2 then y=2,-2. I just wasn’t sure about the complete process to get there. Thanks! $\endgroup$ – Harry Stuart May 30 '18 at 5:27
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You are definitely on the right track! Yes, you should get $y=x$. Just plug it into the original equation and solve (because now you have two conditions that the point should satisfy: the equation and its derivative):

$x^2-2x^2+2x^2=4$
$x=\pm 2$ --> $y=\pm2$

Final points: $(-2,-2), (2,2)$

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$$x^2-2xy+2y^2=4$$ $$\frac{d}{dx}(x^2-2xy+2y^2)=\frac{d}{dy}(4)$$ $$2x-2(y+x\frac{dy}{dx})+4y\frac{dy}{dx}=0$$ $$2x-2y-2x\frac{dy}{dx}+4y\frac{dy}{dx}=0$$ $$\frac{dy}{dx}=\frac{y-x}{2y-x}$$ Now equate $\frac{dy}{dx}=0$ $$\frac{y-x}{2y-x}=0$$ $$x=y,x=2y$$ Solve by substitution, I substituted $y=x$ in $x^2-2xy+2y^2=4$ and got $$x^2-2x^2+2x^2=4$$ $$x=-2,2$$ Now substitute each $x$ value in $x^2-2xy+2y^2=4$ and we get$$y=-2,2$$

The points are $(-2,-2),(2,2)$

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