I don't believe that this problem has a neat solution. I tried it for the parabola , where the general point has parametric form . The normal at this point has equation , and it meets the parabola at the point with parameter . So if d is the length of the chord then , which works out as

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Differentiate that to find that the turning point (which must be a minimum) occurs when . You can factorise that as . The second factor is a quadratic in , and the only positive root is . Therefore the minimum occurs when . If you now substitute the value for into the formula for , taking a=1/4, and finally take the square root to get d, then you will have the length of the shortest chord.