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anonymous
 one year ago
I need help with this question:
Find all the xcoordinates of the points on the curve x^2y^2+xy=2 where the slope of the tangent line is −1.
anonymous
 one year ago
I need help with this question: Find all the xcoordinates of the points on the curve x^2y^2+xy=2 where the slope of the tangent line is −1.

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I've used implicit differentiation to get \[2xy^2+x^22yy'+y+xy=0\] I then substituted y' with 1 and got \[2xy^2x^22y+yx\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I don't know what to do next...

freckles
 one year ago
Best ResponseYou've already chosen the best response.5so you have \[2xy^2+x^22yy'+y+xy'=0 \\ \text{ okay and you replaced } y' \text{ with } 1 \\ 2xy^22x^2y+yx=0 \\ 2xy(yx)+1(yx)=0 \\ (yx)(2xy+1)=0 \\ \text{ so we have } y=x \text{ or } xy=\frac{1}{2} \\ \text{ so try plugging both of these possibilities back in the original equation }\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So wherever I have a y I replace with x, and wherever I have xy I replace with 1/2 ?

freckles
 one year ago
Best ResponseYou've already chosen the best response.5yeah we will see one equation is never true

freckles
 one year ago
Best ResponseYou've already chosen the best response.5and you will see one equation is true for 2 real values

freckles
 one year ago
Best ResponseYou've already chosen the best response.5for example with the xy=1/2 sub \[(xy)^2=(\frac{1}{2})^2 \implies x^2y^2=\frac{1}{4} \\ \text{ but } \frac{1}{4}+\frac{1}{2} \text{ is never } 2 \]

freckles
 one year ago
Best ResponseYou've already chosen the best response.5you can solve the one with y=x pretty easily

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Hmmm... but replacing y with x I get ... x^4 + x^2 = 2

freckles
 one year ago
Best ResponseYou've already chosen the best response.5right that is a quadratic in terms of x^2

freckles
 one year ago
Best ResponseYou've already chosen the best response.5do you know how to solve u^2+u=2?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yes! Oh so now I see it

freckles
 one year ago
Best ResponseYou've already chosen the best response.5I replaced x^2 with u so we can solve for u then replace u with x^2 and then solve for x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ah thank you so much @freckles !!! I got 1, and 1 and it's correct!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I appreciate the help! ^.^
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