The focus of parabola y = - x square + K and X axis is a (a, 0), B (B, 0). If a square + b square = 4, find the value of K

The focus of parabola y = - x square + K and X axis is a (a, 0), B (B, 0). If a square + b square = 4, find the value of K

-2
Combine the equations of the two focal points to get the result
0 = A & sup2; + K, 0 = B & sup2; + K, and a & sup2; + B & sup2; = 4

The known parabola y = x square + (2k + 1) x-k square + k
(1) It is proved that there are two intersections between the parabola and the x-axis;
(2) When k = - 1, find the coordinates of the intersection of the parabola and the coordinate axis

(1) Discriminant
Δ=b²-4ac
=(2k+1)²-4(-k²+k)
=4k²+4k+1+4k²-4k
=8k²+1
Because 8K & # 178; ≥ 0, 8K & # 178; + 1 ≥ 1 > 0
X & # 178; + (2k + 1) x-k & # 178; + k = 0 must have two different points, so this parabola has two intersections with the X axis
（2）
When k = - 1,
Parabola is y = x & # 178; - X-2
When y = 0,
x²-x-2=0
(x-2)(x+1)=0
x1=2
x2=-1
So the coordinates of the intersection point with the X axis are (2,0) (- 1,0)
At the same time, when x = 0, y = - 2, so the coordinates of the intersection point with y axis (0, - 2)

Known parabola y = x ^ 2 - (K + 3) x + 2k-1!
It is known that the parabola y = x ^ 2 - (K + 3) x + 2k-1
Q: let the parabola and X-axis intersect at two points a and B (a is on the left side of B), the vertex is C, and the ordinate of C is m. how to find the value of AB ^ 2 / M?

Let a (x1,0) and B (x2,0) be derived from the known conditions: C ((K + 3) / 2, m) substitute the point C coordinate into the equation to get: [(K + 3) / 2] ^ 2 - (K + 3) * (K + 3) / 2 + 2k-1 = m, and simplify it to: (K + 3) ^ 2-8k + 4 = - 4mab ^ 2 = (x2-x1) ^ 2 = (x1 + x2) ^ 2-4x1x2 = (K + 3) ^ 2-4 (2k-1) = (K + 3) ^ 2-8k + 4 = - 4m, so AB ^ 2 / M = - 4m

It is known that the parabola y = ax ^ 2 + BX + C intersects with the straight line y = KX + 4 at two points a (1, m), B (4, 8), and intersects with the X axis at the origin O and point C
1. Finding the corresponding function analytic expression of straight line and parabola
2. Is there a point D on the parabola above the x-axis, such that s triangle OCD = 1 / 2S triangle OCB? If so, request d that meets the condition. If not, explain the reason

Because the parabola passes through the origin, C = 0 (C is the intercept, that is, the distance from the intersection of the parabola and the y-axis to the origin), so the parabola is y = ax ^ 2 + BX. Take point B into the first-order function, solve y = x + 4, and then take point a into it, you can get the coordinates of a as (4,8), and then take a, B into