Anyone know this formula?

[racuna]

Hi all ,

I have attampted this type of problem before, but every time I do I just keep getting more and more confused.

If I have a problem that says:
Find the point(s) on the parabola y=2x^2-1 that are closest to the point (0,1)
what is the formula that I need?

I think I start with a general distance formula, but I don't understand the rest.
Can anyone please clarify the steps to this problem?

P.S. Does this kind of problem only deal with parabolas?

Thanks,
Rachel

 

[stapel]

Find the point(s) on the parabola y=2x^2-1 that are closest to the point (0,1)

Plug (0, 1) and (x, 2x² - 1) into the Distance Formula. Minimize by the usual methods (first derivative, critical points, etc).

Does this kind of problem only deal with parabolas?

No. Any curve will do.

Eliz.

 

[pka]

Suppose that (p,q) is a point on the parabola. Then q=2p<SUP>2</SUP>−1.

The distance from (p,q) to (0,1) is \(\displaystyle \L
d = \sqrt {p^2 + \left( {q - 1} \right)^2 }\).

So that \(\displaystyle \L
d = \sqrt {p^2 + \left( {2p^2 - 2} \right)^2 }\) or \(\displaystyle \L
d' = \frac{{2p + 2(2p^2 - 2)(2p)}}{{2\sqrt {p^2 + \left( {2p^2 - 2} \right)^2 } }}\).

Now find where the numerator is zero.