I asked ChatGPT to generate a question on conic sections, it gave this:
A line with slope 2 intersects a parabola with vertex at (0,0) at exactly one point. Determine the equation of the parabola.
When queried further, it elaborated that the line $$y=2x$$ and the parabola $$y=\frac18x^2$$ intersect at only one point. But, clearly, they intersect more than once. In general, if a line of given slope intersects a parabola at only point, then does it fix the parabola?
A general equation of a parabola with vertex at the origin is $y=ax^2$. A line with the slope $2$ has equation $y=2x+c$. Let $x=x_0$ is the point of intersection. SInce the requirement is for line to be tangent, the slope of the line should be equal to derivative at that point. Thus, $2ax_0=2$ or $x_0=\frac{1}{a}$. Which means that for any value of $a$ we can find a point on the parabola thru which a tangent line can be drawn with the slope of $2$. The equation of the line will be $y=2x-\frac{1}{a}$. The problem does not have a single solution.
For $y=\frac{x^2}{8}$, the tangent line with the slope $2$ will be $y=2x-8$ and the point of tangency is $(8,8)$.
