Let P be a moving point on the curve Y 2 = 4x, then the minimum value of the sum of the distance from point P to point A (-1, 1) and the distance from point P to line X =- 1.

From y 2 = 4x = 2px, we get p = 2, p/2 = 1, so the focus is F( 1, 0) and the directrix x=-p/2=- 1.

Let p be a vertical line with PN x=- 1 According to the definition of parabola,

The distance from a point on a parabola to a straight line is equal to the distance from the focus.

So there is |PN|=|PF|, which connects two points, F and A, and the shortest line segment between the two points is |FA|≤|PA|+|PF|.

So p is the intersection of AF and parabola, the distance from point P to point A (-1, 1) and the distance from point P to a straight line X=- 1.

The minimum value of the sum of distances is | fa | = √ (12+2 2) = √ 5.

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