If the distance between a point m on the parabola y ^ 2 = 3x and the coordinate origin o is 2, then the distance between the point m and the focus of the parabola is 2_____ .

Let the coordinates of M be (x0, Y0) and let X02 + Y02 = 22 (1)
y02=3x0 (2)
Substituting (2) into (1) yields X02 + 3x0 = 4, i.e
The solution is x0 = - 4 (rounding off) or x0 = 1
From the parabolic equation, we know that P = 3 / 2, so p / 2 = 3 / 4, so the Quasilinear equation of parabola x = - 3 / 4
From the definition of parabola, ∣ MF ∣ = x0 - (- 3 / 4) = 1 + 3 / 4 = 7 / 4

It is known that the vertex of the parabola Ω is the coordinate origin o, the focus f is on the positive half axis of the y-axis, and the line L passing through the point F intersects the parabola at two points m and n,
And satisfy the vector om · vector on = - 3
Solving the equation of parabola Ω

After specialization to path, the two intersections are (- P, P / 2), (P, P / 2)
So quantity product = - 3P ^ 2 / 4
If the direct simultaneous solution is also possible, use the Veda theorem to find out x1x2, y1y2, and then substitute om * on = - 3
If the multiple choice questions are directly specialized.

How many units can the parabola y = x square be translated downward so that the area of the triangle bounded by the vertex of the parabola and its intersection with the coordinate axis is 8

Let m units be shifted downward
The vertex coordinate is C (0, - M), and its intersection with the coordinate axis is a (√ m, 0) B (- √ m, 0) AB = 2 √ M
The area of a triangle is 8
1/2AB×OC=8
1/2×2√m×m=8
(√m)^3=8
√m=2
m=4
Parabola y = x square translation 4 units down
Good luck and progress in your study

What is the area of the triangle formed by the intersection of the parabola y = - x ^ 2-2x + 3 and the coordinate axis______ .

To draw a picture, first find two zeros (- 3,0), (1,0), and the image intersects with the Y axis at (0,3), so the triangle area is 1 / 2 × (1 + 3) × 3 = 6

If the distance between a point m on the parabola y ^ 2 = 3x and the coordinate origin o is 2, then the distance between the point m and the focus of the parabola is 2_____ .