Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$x^2 + xy + y^2 = 7$

$x$-axis = $\frac{\sqrt{21}}{3}$, $\frac{-2\sqrt{21}}{3}$

I don't understand how to find the $y$-axis.

share|improve this question
    
Isn't the question asking you to find where the tangent to the curve $x^2+xy+y^2=7$ is horizontal and vertical? –  Joe Johnson 126 Nov 14 '12 at 12:54
    
If so, you can use implicit differentiation. –  Joe Johnson 126 Nov 14 '12 at 12:55
    
I tried that and got -2 sqrt 21/ 3, sqrt 21/ 3, which is the opposite of the x axis. Would that be correct? –  Courtney Nov 14 '12 at 13:00
1  
Note that our curve function is symmetric in $x$ and $y$. By symmetry, for parallel to $y$-axis we interchange the roles of $x$ and $y$. Thus (i) your computation is correct and (ii) you need not have computed. If we did not have symmetry, the $x$-axis argument could be imitated by finding $\frac{dx}{dy}$. –  André Nicolas Nov 14 '12 at 13:04
add comment

1 Answer

Usually for these problems one uses the implicit derivative, which for this problem is $$y'=\frac{-2x-y}{x+2y}.$$ Then horizontal tangents occur when the top is zero, i.e. $2x+y=0$, and vertical tangents occur when the bottom is zero, i.e. $x+2y=0$. These equations are then plugged into the original relation $x^2+xy+y^2=7$ to get the actual coordinates of the points.

Note: Just noticed that Joe Johnson made this same suggestion re. implicit derivative!

EDIT: I got the horizontal tangents occur at $(x,y)=(+\sqrt{7/3},-2\sqrt{7/3})$ and at $(x,y)=(-\sqrt{7/3},+2\sqrt{7/3}).$ The vertical tangent points were like these, only switch the ordering of the pairs $(x,y)$. Maybe because the original ellipse $x^2+xy+y^2=7$ has its major axis at 45 degrees rotated.

share|improve this answer
    
Thank you very much! It makes much more sense now! :) –  Courtney Nov 14 '12 at 13:07
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.