If the distance from a point P to the x-axis on the parabola y ^ 2 = 4x is 6, then the distance from point P to the focus is 0

The guide line is x = - 1
Distance from point P to focus = distance from point P to guide line = 6 + 1 = 7

If a point P on the parabola y ^ 2 = 4x has the smallest sum of its distance to the point m (4,2) and its distance to the focus F of the parabola, what is the coordinate of point P?
I hope you can be more specific~

Make the directrix of the ellipse
It is known from the second definition that the distance from P to the guide line is equal to the distance from the focus
When PM is parallel to the x-axis, the sum of the distances from P to the parabolic focus f is the smallest
So the ordinate of P is 2 (same as m)
Substituting y ^ 2 = 4x, then x = 1
So p (1,2)

Given that the vertex of the parabola y = (x + a) square + 2A square + 3a-5 is on the coordinate axis, find the value of the letter A, and point out the vertex coordinates

The vertex of parabola y = (x + a) square + 2A square + 3a-5 is on the coordinate axis, that is, when x = - A, y = 2A ^ 2 + 3a-5 = 0, the solution is a = 1 or a = - 5 / 2, and the vertex coordinate is (- 1,0) or (5 / 2,0)

Given that the vertex of the parabola y = (x + a) square + 2A square + 3a-5 is on the coordinate axis, find the value of the letter A, and point out the vertex coordinates

y=(x+a)^2+2a^2+3a-5
If the vertex is on the X axis, 2A ^ 2 + 3a-5 = 0
(a-1)(2a+5)=0
A = 1 or a = - 5 / 2
If the vertex is on the Y axis, x = - a = 0
a=0
A = 0 or a = 1 or a = - 5 / 2

If the distance from a point P to the x-axis on the parabola y ^ 2 = 4x is 6, then the distance from point P to the focus is 0