According to the Kepler's first law, the orbit of an asteroid would be an ellipse having the planet at one of the focuses (to a good approximation). Then how come can it ever fall to the planet?

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    $\begingroup$ consider that the radius of the perigee may be less than the radius of the planet $\endgroup$ – Jim Jul 11 '16 at 13:22
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    $\begingroup$ An asteroid is usually not in orbit around the planet it impacts. $\endgroup$ – Stéphane Rollandin Jul 11 '16 at 15:36

Most asteroids are in an elliptical orbit around the Sun in the inner Solar System, i.e. a region comprising Mercury, Venus, Earth, Mars and the Asteroid Belt. What can happen is that an asteroid's elliptical orbit intersects a planet's orbit and this might gives rise to a collision.

Most of times, when an asteroid gets too close to a planet, it has too much energy to be captured, so it does an hyperbolic orbit around the planet. The focus of this hyperbola is the center of the planet. So if the maximum approach of the asteroid relative to that focus is less than or equal to the radius of the planet, then they are going to collide, as in the figure bellow.

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If the asteroid losses part of its energy by transferring it to other planets before approaching a particular planet, then it might be captured in an elliptical orbit. But even in this case it collides with the planet if the maximum approach to the focus is less than or equal to the radius of the planet.


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