Given that two points a and B on the parabola y = 2x2 form an isosceles right triangle with the origin o, the coordinates of two points a and B are obtained

Let a (a, 2A2), then | a | = 2A2, ∵ a ≠ 0, | a = ± 12, ∵ a (12, 12), B (- 12, 12)

If we know that a straight line L: y = K (x + 1), a parabola C: Y & # 178; = 4x, then how many lines l have a common point with C?

Three, k = ± 1, is tangent, k = 0, is axis of symmetry;
The three linear equations are: y = x + 1, y = - X-1, y = 0

Given that the coordinate of the intersection of the parabola y = x & # 178; - 4x + m / 2 and the X axis is (1,0), then the coordinate of the other intersection of the parabola and the X axis is?

3.0

If the parabola y = - X & # 178; + 2x + m + 1 intersects the x-axis with two points a and B, and both points a and B are on the positive half axis of the x-axis, the value range of M is obtained (no graph ~)

Parabola y = - X & # 178; + 2x + m + 1, opening downward, axis of symmetry x = 1
It is necessary to satisfy that the intersection points a and B with the x-axis are in the positive half axis
There are two
B ^ 2-4ac = 4-4 * (- 1) * (M + 1) = 4 + 4 (M + 1) = 4m + 8 > = 0, M > = - 2
② The product of two is greater than 0
X1 * x2 = - (M + 1) > 0, M is obtained

If the parabola y = - 3x & # 178; + 2x + C intersects the X axis at two points, then the value range of C is

If there are two intersections between the parabola and the x-axis, then B & # 178; - 4ac > 0, so 4 + 12C > 0, C > 1 / 3

It is known that the intersection of the parabola y = AX2 + 4ax + T and the x-axis is a (- 1, 0); (1) find the coordinates of the other intersection B of the parabola and the x-axis; (2) d is the intersection of the parabola and the y-axis, C is a point on the parabola, and the area of the trapezoid ABCD with ab as the bottom is 9. Find the analytical formula of the parabola; (3) e is the point in the second quadrant with the distance ratio of 5:2 to the x-axis and y-axis Point E is on the parabola in (2), and it is on the same side of the symmetry axis of the parabola as point A. question: is there a point P on the symmetry axis of the parabola to minimize the perimeter of △ ape? If it exists, find out the coordinates of point p; if not, explain the reason

(1) The axis of symmetry of parabola is x = - 2, ∵ points a and B must be symmetric about the axis of symmetry, ∵ the other intersection is B (- 3, 0). (2) ∵ the coordinates of a and B are (- 1, 0), (- 3, 0), ∵ AB = 2, ∵ the axis of symmetry is x = - 2, ∵ CD = 4; let H. ∵ s trapezoid ABCD = 12 × (2 + 4) H = 9, ∵ H = 3, that is | - t | = 3, ∵ t = ± 3, when t = 3, substitute (- 1, 0) into the solution When t = - 3, substituting (- 1, 0) into the analytic expression, we can get a = - 1, a = 1 or a = - 1, and the analytic expression is y = x2 + 4x + 3 or y = - x2-4x-3; (3) according to the meaning of the title, e is on y = - 52X, and on the right side of x = - 2, together with the parabola y = x2 + 4x + 3, we can get x2 + 132x + 3 = 0, x = - 6 or x = - 12 ∵ E and point a are on the same side of the parabola symmetry axis, ∵ e (- 12, 54). A is on the opposite side The symmetry point B (- 3,0) of the axis is called. The equation connecting the symmetry axis of B and e to the point P, ∵ be is y − 054 − 0 = x + 3 − 12 + 3, that is, y = 12, that is, P (- 2,12) when y = 12x + 32, ∵ x = - 2. When y = - 52X and y = - x2-4x-3 are combined, we can get x2 + 32x + 3 = 0. This equation has no solution. In conclusion, there is a point P (- 2,12) on the symmetry axis of the parabola, which minimizes the perimeter of △ ape

A point of intersection of parabola y = ax + 4ax + T and X axis is a (- 1,0)
1) Find the coordinates of another intersection B of parabola and X-axis
2) D is the intersection of the parabola and the Y axis, C is a point on the parabola, ab ‖ CD, the area is 9, find the analytical formula of the parabola

Solution 1) is rooted in the coefficient relation
x1+x2=-4
X1 = - 1, so x2 = - 3
The other intersection is (- 3,0)
2) D (O, t)
Because AB is parallel to CD
So let C (x, 0)
The area of that triangle is 9
It's ABC!
So ABC = | t | * (- x) / 2 = 9
And C is on a parabola
So it was brought into Fang Chengzhong
The analytical formula of parabola can be obtained by solving the two equations simultaneously

If the two intersection points of the parabola y = x ^ 2-mx + m-2 and the X axis are on both sides of the origin, then the value range of M

The two intersections are on both sides of the origin, and the coefficient of x ^ 2 is greater than 0, and the opening is upward
So the intersection point of X and y must be below x, that is, y when x = 0

We know the square of parabola y = x-mx + 2m-4
When the parabola and X-axis intersect at two points a and B (point a is on the left side of y-axis, point B is on the right side of y-axis), and the length ratio of OA to ob is 2:1, the value of M is obtained

A:
y=x^2-mx+2m-4
=(X-2) [x - (m-2)] has two intersections with the X axis,
x1=2,x2=m-2
Point B is (2,0) and point a is (m-2,0)
And m-2

If the image of the function y = (M2-4) X4 + (m-2) X2 is a parabola with the vertex at the origin and the axis of symmetry is y-axis, then M=______ ．

The graph of ∵ function y = (M2-4) X4 + (m-2) X2 is the vertex at the origin, ∵ 4ac − b24a = 0, ∵ M = ± 2, and ∵ symmetry axis is Y axis, ∵ m ≠ 2, ∵ M = - 2

Given that two points a and B on the parabola y = 2x2 form an isosceles right triangle with the origin o, the coordinates of two points a and B are obtained