The equation x & sup2; + Y & sup2; + ax + 2ay + 2A & sup2; + A-1 = 0 indicates the value range of garden rule a

The equation x & sup2; + Y & sup2; + ax + 2ay + 2A & sup2; + A-1 = 0 indicates the value range of garden rule a

x²+y²+ax+2ay+2a²+a-1=0
(x+a/2)²+(y+a)²=-3a²/4-a+1
If R & sup2; = - 3A & sup2 / / 4-A + 1 > 0
3a²+4a-4

The straight line passing through point a (1,0) intersects with circle C: x ^ 2 y ^ 2-6x-8y 9 = 0 at two points P and Q, and intersects with straight line L: X 2Y 4 = 0 at point n. if the midpoint of PQ is m, it is proved; | am | * | an | is the fixed value

X^2+Y^2-6X-8Y+9=0
（x-3)^2+(y-4)^2=16
The center C coordinates are (3,4)
Straight line L: x + 2Y + 4 = 0
This kind of problem, if there is no clever method, is torture to junior high school students
Am and an in two similar triangles, based on this idea, only connect the center of the circle and point a, extend to the known line L, the intersection point is p
The straight line passing through the center of the circle C (3,4) and a (1,0) is: y = 2x-2
It is perpendicular to the line L. the perpendicular foot is the intersection P. the distance CP = Ca + AP from the center of the circle to the line L is a fixed value, and the triangle AMC is similar to the triangle ANP, an: AP = AC: am
An * am = AP * AC is the fixed value