It is known that the collimator of parabola C: y ^ 2 = 4x and X-axis intersect at point m, the straight line L with slope k of point m intersects with parabola C at two points ab 1 F is the focal point of parabola C. if the module am = 5 / 4, the module AF can find the value of K 2 whether there is such a K, so that there is always Q on the parabola C, and if QA vertical QB exists, ask for the value range of K

It is known that the collimator of parabola C: y ^ 2 = 4x and X-axis intersect at point m, the straight line L with slope k of point m intersects with parabola C at two points ab 1 F is the focal point of parabola C. if the module am = 5 / 4, the module AF can find the value of K 2 whether there is such a K, so that there is always Q on the parabola C, and if QA vertical QB exists, ask for the value range of K

(1) In AMH, we make a triangle of ah perpendicular to x-axis
|MH | = distance from a to the guide line = | AF|
|If MH | / | am | = 4 / 5, k = tanamh = 3 / 4
(2) Note a (x1, Y1) B (X2, Y2) Q (A & sup2;, 2a)
Y = K (x + 1) and parabolic equation
x1+x2=(4-2k²)/k²
x1x2=1
y1+y2=4/k
y1y2=4
Vector QA = (x1-a & sup2;, y1-2a)
Vector QB = (x2-a & sup2;, y2-2a)
By QA * QB = 0
(a²+5)(a²+1)k²-8ak-4a²=0
We obtain k = - 2A / (A & sup2; + 5) or 2A / (A & sup2; + 1)
-2a/(a²+5)≥-√5/5
2a/(a²+1)≤1
So - √ 5 / 5 ≤ K ≤ 1 and K ≠ 0