Question 1024076
Since the tangent line to the ellipse passes through the origin, the line will have the form y = mx, where m is the slope of the line.

==> {{{x^2+3(mx)^2-x+2mx=0}}}, after substitution

==> {{{(1+3m^2)x^2+(2m-1)x = 0}}} after simplifying...

For tangency, the discriminant {{{b^2 - 4ac = 0}}}

==> {{{(2m-1)^2 - 4(1+3m^2)(0) = 0}}}, or

{{{(2m-1)^2 = 0}}}, or m = -1/2.

Therefore the line tangent to the ellipse at (0,0) is y = -x/2.