∵ quadrilateral ABCD is a square,
∴AC⊥BD,
∵PD⊥ plane ABCD, AC∈ plane ABCD,
∴AC⊥PD,
∫PD∩BD = D,
∴AC⊥ aircraft PDB,
∫AC∈ plane ACE,
Aircraft, ACE⊥. Aircraft.
2. link EO,
To sum up, the AC(O)⊥ plane PDB,
∴ < AEO is the angle formed by AE and plane PDB,
AB = AD = PD = a,
∴BD=√2a,
PB=√(a^2+2a^2)=√3a,
BE=BP/3=√3a/3,
On the plane PBD, EF⊥BD, vertical foot h,
EH/PD=BE/PB= 1/3,
∴EF=a/3,
OB=BD/2=√2a/2,
BH/BD=BE/PB= 1/3,
∴BH=√2a/3,
OH=OB-BH=√2a/6,
OE=√(OF^2+EF^2)=√6a/6,
ae=√(ao^2+oe^2)=√(2a^2/4+6a^2/36)=√6a/3,
∴cos<; AEO = EO/AE =(√6a/6)/(√6a/3)= 1/2
The angle between AE and PDB plane is 60 degrees.
3.∫DF// plane ACE,
And plane DFB∩ plane ACE=OE,
∴DF//OE,
O is the midpoint of BD,
∴OE is the center line of △BDF,
∴EF=BE=PB/3=√3a/3,
∴pf=pd-ef-be=pd-2be=√3a-2√3a/3=√3a/3.
Conic curve formula: ellipse
1, elliptic standard equation with the center at the origin and the focus at the X axis: where x? /a? +y? /b? = 1,