A planet travels in an ellipse around the origin with the sun close to the origin. The planet's position in the ellipse is given by:

$x(t)=a\cos(kt)$

$y(t)=b\sin(kt)$

where $a>b$ and $k$ are constants and $0 ≤ t ≤ 2\pi$ ($t$ is a time variable).

The sun's position is given by $(x,y) = ( \sqrt{ a^2 − b^2}, 0)$.

I've learnt that taking the derivative of the functions for position twice gives the functions for the acceleration. How would this look and how can I compare this answer with the functions for positions to show that the vector of acceleration is always pointed towards the origin?

In my attempt I've got that the acceleration is

$$a_x(t) = −ak^2cos(kt), a_y(t) = −bk^2sin(kt)$$

So the difference between position and acceleration is $−k^2$, provided that I'm correct so far. However I do not understand how this would show that the acceleration always is pointed towards the origin. Does the place of the sun even have any relevance? Considering that no masses and gravitation is taken account for.