## calyne 3 years ago The equation x^2 - xy + y^2 = 3 represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.

1. dpaInc

to find the x-intercepts plug in y=0 and solve for x. to show that the tangent lines are parallel, plug in the coordinates of these points to the derivative y'. if they're parallel, they should have the same value...

2. calyne

wait so the x-intercepts are [+-]sqrt(3)?

3. dpaInc

yes

4. calyne

well i got y' = -2x/(-1+2y)

5. calyne

so what, -2*sqrt(3)/(-1+0) = 2*sqrt(3), and then the same for -sqrt(3)..? what's that then? -2*-sqrt(3)/(-1+0) = -2sqrt3 i don't get it

6. dpaInc

check your derivative... again... i got something different

7. calyne

oh right crap

8. calyne

ok y' = (-2x+y)/(-x+2y) ?

9. dpaInc

that's what i got..

10. calyne

so substituting (sqrt(3),0) m=2?

11. dpaInc

yep. notice you get the same thing for the other point... so they're parallel.

12. calyne

yup. gotcha. thanks

13. dpaInc

yw