What normal to the curve y = x^2 forms the shortest chord?
This is what I did:
I used parametric form of parabola and did stuff, put it in distance formula.. d/dt = 0 it, I'm getting a complicated polynomial ( 7th degree ) after doing that..
What normal to the curve y = x^2 forms the shortest chord?
This is what I did:
I used parametric form of parabola and did stuff, put it in distance formula.. d/dt = 0 it, I'm getting a complicated polynomial ( 7th degree ) after doing that..
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.