Parabola y = - X & # 178; + [M-1] x + m intersects with y axis at [0,3]. Find its intersection with X axis and coordinates of parabola vertex

Parabola y = - X & # 178; + [M-1] x + m intersects with y axis at [0,3]. Find its intersection with X axis and coordinates of parabola vertex

Substituting (0,3) into the equation, we can get 3 = M = = > the parabolic equation is: y = - x ^ 2 + 2x + 3, from - x ^ 2 + 2x + 3 = 0 = = > (x-3) (x + 1) = 0 = = > x = - 1 or x = 3, the intersection point with X axis is (- 1,0) and (3,0), and y = - x ^ 2 + 2x + 3 = - (x-1) ^ 2 + 4, so

If the distance between a point P and the focus f on the square of the parabola y = 1 / 4x is 5, then the coordinates of point P are

First, the parabola y = 1 / 4x is squared to the standard form x2 = 4Y
Let P (m, n)
The distance from P to focus f = the distance from P to the guide line
The distance from P to the guide line = n - (- 1) = 5
So n = 4
Then we take P (m, 4) into the square of the parabola y = 1 / 4x
M = ± 4
So p (± 4,4)

The square of parabola y = - 4x, the distance from a point to the focus is 4, then its coordinate is

The guide line is x = 1
Defined by parabola
The distance to focus is equal to the distance to guide line
Let (a, b)
Distance to guide line = | A-1 | = 4
Opening to the left, so a