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 Post subject: Minimum distancePosted: Fri, 23 May 2003 18:45:21 UTC
the point M is on the parabola x^2=64y.Which is the minimum distance between M and the line 3x+4y+37=0?

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 Post subject: Posted: Fri, 23 May 2003 20:03:04 UTC
 Member

Joined: Fri, 23 May 2003 19:49:36 UTC
Posts: 11
Location: Brasil
I would do it like this...

M is on x^2=64y, or y=(x/8)^2. So, M coordinates have the form (x, (x/8)^2).

Distance from M to given line is:

D = |3x + 4(x/8)^2 + 37| / sqrt (9 + 16)

D^2 = (3x + x^2/16 + 37)^2 / 25 (let's call D^2 = W)

If D is minimum, then D^2 = W is also minimum (remember that D>=0).

dW/dx = (1/25) (2(3x + x^2/16 + 37) (3 + x/8)

W is minimum when dW/dx = 0:

(1/25) (2(3x + x^2/16 + 37) (3 + x/8) = 0

So, either

(3x + x^2/16 + 37) = 0 -> no real roots

or

(3 + x/8) = 0 -> x = -24

Putting x = -24 in the coordinates of M, we have the answer

M(-24, 9)

_________________
ViniÂ®

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 Post subject: Posted: Fri, 23 May 2003 21:45:07 UTC
Good job,i am checking it on the paper now.Thx!

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