Question

A comet travels along a parabolic path around the Sun. The Sun is the focus of the path. When the comet is at the vertex of the path, it is 60,000,000 kilometers from the Sun. Write an equation that represents the path of the comet. Assume the focus is on the positive y-axis and the vertex is (0, 0)(0, 0).

Answer 1

Remember that for a function of the form $y=a x^{2}$, a parabola, the focus will be: $(0, \frac{1}{4a} )$.
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 $a$ using the focus of the parabola:
$\frac{1}{4a} =60'000.000$
$1=(60'000.000)(4a)$
$1=240'000.000a$
$a= \frac{1}{240'000.000}$

Now, the only thing left is replacing the value of $a$ in our equation $y=a x^{2}$:
$y=( \frac{1}{240'000.000)} ) x^{2}$
$y= \frac{ x^{2} }{240'000.000}$
 
We can conclude that the equation that represents the path of the comet is $y= \frac{ x^{2} }{240'000.000}$