P(4,2)是圆C:x^2+y^2-24x-28y-36=0内的一点,圆上的动点A,B满足∠APB=90°
Q(x,y)
2x=xA+xB,2y=yA+yB
4x^2=(xA+xB)^2
4y^2=(yA^2+yB)^2
x^2+y^2-24x-28y-36=0
(xA)^2+(yA)^2-24xA-28yA-36=0
(xB)^2+(yB)^2-24xB-28yB-36=0
[(xA)^2+(yA)^2-24xA-28yA-36]+[(xB)^2+(yB)^2-24xB-28yB-36]=0
(xA)^2+(xB)^2+(yA)^2+(yB)^2-24*(xA+xB)-28*(yA+yB)-72=0
(xA+xB)^2-2xA*xB+(yA+yB)^2-2yA*yB-24*2x-28*2y-72=0
4x^2+4y^2-48x-56y-72=2(xA*xB+yA*yB)
PA⊥PB
[(yA-2)/(xA-4)]*[(yB-2)/(xB-4)]=-1
(xA-4)*(xB-4)+(yA-2)*(yB-2)=0
xA*xB+yA*yB=4(xA+xB)+2(yA+yB)-20
xA*xB+yA*yB=4*2x+2*2y-20
16x+8y-40=2(xA*xB+yA*yB)
4x^2+4y^2-48x-56y-72=16x+8y-40
x^2+y^2-16x-16y-8=0
AB中点Q的轨迹方程是园:(x-8)^2+(y-8)^2=136