A parabola passes through the origin, and the vertex coordinates (- 2,4) Then the analytical expression of this parabola is? RT can set this analytic expression as y = a (x + 2) + 4, but the title says that we have to go through the origin, that is to say, C = 0, but let C = 4 in the vertex expression. Isn't this contradictory? I just can't understand it here

A parabola passes through the origin, and the vertex coordinates (- 2,4) Then the analytical expression of this parabola is? RT can set this analytic expression as y = a (x + 2) + 4, but the title says that we have to go through the origin, that is to say, C = 0, but let C = 4 in the vertex expression. Isn't this contradictory? I just can't understand it here

Let the parabolic equation be
y=a(x+2)^2+4
=ax^2+2ax+4a+4
Because of crossing the origin
So 4A + 4 = 0
a=-1
arcsinx-x
y=-(x+2)^2+4

If we know that the vertices a and B of an equilateral triangle AOB are on the parabola y2 = 6x, and O is the coordinate origin, then the side length of △ AOB = ()
A. 123B. 63C. 363D. 243

From the symmetry of parabola, we can get the equation of ∠ AOX = 30 °, the equation of straight line OA is y = 33x, simultaneous y = 33xy2 = 6x, and the solution is a (18, 63); | ab | = 123

One vertex of an equilateral triangle is at the origin, and the other two vertices are on the parabola y ^ 2 = 2px,
One vertex of an equilateral triangle is at the origin, and the other two vertices are on the parabola y ^ 2 = 2px
Side length, the specific process,

Obviously the other two points are symmetric about the x-axis
So the angle between the two sides passing through the origin and the x-axis is 30 degrees
k=tan30=√3/3
The straight line is y = √ 3x / 3
y²=2px
So x & # / 3 = 2px
x=6p
y=√3x/3=2√3p
That is, the point of intersection with parabola is (6p, ± 2 √ 3P)
Their distance is the side length
So the side length is 2 √ 3P - (- 2 √ 3P) = 4 √ 3p

If the equilateral triangle MNP is inscribed on the parabola y ^ 2 = 2x and the vertex is at the origin, then the area of the triangle is?
I want more solutions

According to the fact that the parabola is symmetrical about the x-axis and the triangle MNP is an equilateral triangle, we can see that one of the three points of MNP coincides with the origin. Let the other two points be (a, b)
(a, - b) and (a > 0)
B ^ 2 = 2A (point on parabola)
(2b) ^ 2 = a ^ 2 + B ^ 2 (equal sides)
The solution is a = 6
The area of triangle is the absolute value of ab
Do it yourself