Christmas Streak 43/88: Minimize This Please
I drew the graph of y = 4 1 x 2 , and then picked two points A = ( 0 , 1 ) and B = ( 5 , 2 0 1 7 ) .
Now I'm going to pick a point P on the graph such that A P + B P is minimized.
What's the y -coordinate of P ?
This problem is a part of <Christmas Streak 2017> series .
The answer is 6.25.
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1 solution
But A(1,0) is NOT the focus of y = 4 1 x 2 . The focus is (0, 1).
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Thanks... I had a headache yesterday and wasn't thinking straight ;;>_>
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But in this case, the answer is found by intersecting the line connecting (0, 1) and (5, 3) with the parabola. So y = 4 1 x 2 = 1 + 0 . 4 x , resulting in x = 2 . 1 5 4 0 6 5 9 2 2 8 5 3 8 0 1 6 1 2 5 0 0 2 8 4 1 9 6 6 1 6 1 , and the corresponding y comes to y = 4 1 x 2 = 1 . 1 6 .
@H.M. 유 Why is AP=PH? . y=0 is not the directrix .
Moreover , if AP = PH then AP+BP=BP+PH
It seems to me that AP + BP is found by drawing a straight line from A to B. Where it intersects the graph at (3,2.25) should be point P. The shortest distance between two points is a straight line. Ed Gray
Your solution needs some editing.
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Exactly where do you mean?
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If H is the foot of the perpendicular from P to the x-axis, A P = P H + 1 . Because, the directrix is y = − 1 .
Actually, A ( 0 , 1 ) is the focus of y = 4 1 x 2 .
Namely, if we let H be the foot of perpendicular from P to the line y = − 1 , A P = P H .
Therefore A P + B P = B P + P H .
But then for this to be minimized, we need B P to be perpendicular to the x -axis.
Hence P ( 5 , 4 2 5 ) .