Remember that for a function of the form [tex]y=a x^{2} [/tex], a parabola, the focus will be: [tex](0, \frac{1}{4a} )[/tex].
We know that the vertex of the parabola is (0,0), and the sun is the focus of the parabola; we also know that the focus is in the positive y-axis and when the comet is as its vertex, the distance between them is 60'000.000 km; therefore the focus of our parabola is (0, 60'000.000).
Now that we know the focus, we can find [tex]a[/tex] using the focus of the parabola:
[tex] \frac{1}{4a} =60'000.000[/tex]
[tex]1=(60'000.000)(4a)[/tex]
[tex]1=240'000.000a[/tex]
[tex]a= \frac{1}{240'000.000} [/tex]
Now, the only thing left is replacing the value of [tex]a[/tex] in our equation [tex]y=a x^{2} [/tex]:
[tex]y=( \frac{1}{240'000.000)} ) x^{2} [/tex]
[tex]y= \frac{ x^{2} }{240'000.000} [/tex]
We can conclude that the equation that represents the path of the comet is [tex]y= \frac{ x^{2} }{240'000.000} [/tex]
