Question

请帮助一道函数题

Answer 1

P': y = x^2+mx -2m^2
1.
x^2+mx -2m^2 = 0
△= m^2+8m^2
= 9m^2 > 0
与x-axis 有两个不同的交点

2.
m=3
y = x^2+3x -18
P(0,n), line pass through P and perpendicular to y-axis
l: y = n
l cut P at A and B such that AP=2PB

y = x^2+3x -18
y'= 2x+3 =0
x = -3/2
x = -3/2 cut P' symmetrically

for a>0,

let A(-3/2 - a, n) , then
B(-3/2 + a, n)

x^2+3x-18 = n
x^2+3x-(18+n) = 0
product of roots -a^2+ 9/4 = -(18+n)

AP = 2PB
|-3/2-a| = 2|-3/2+a|
a^2+3a+9/4 = 4a^2-12a+9
3a^2-15a+27/4 =0
4a^2-20a+9=0
(2a-9)(2a-1)=0
a = 1/2 or 9/2

-a^2+9/4 = -(18+n)
n = a^2 -81 /4
= -20 or 0