Question

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.

Answer 1

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...

Answer 2

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

Answer 3

yes

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

oh right crap

Answer 8

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

Answer 9

that's what i got..

Answer 10

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

Answer 11

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

Answer 12

yup. gotcha. thanks

Answer 13

yw