2a^2+3c^2=2b^2+3d^2=(ad-bc)^2=6 求 :(a^2+b^2)(c^2+d^2)

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2a^2+3c^2=2b^2+3d^2=(ad-bc)^2=6求:(a^2+b^2)(c^2+d^2)2a^2+3c^2=2b^2+3d^2=(ad-bc)^2=6求:(a^2+b^2)(c^2+d^

2a^2+3c^2=2b^2+3d^2=(ad-bc)^2=6 求 :(a^2+b^2)(c^2+d^2)
2a^2+3c^2=2b^2+3d^2=(ad-bc)^2=6 求 :(a^2+b^2)(c^2+d^2)

2a^2+3c^2=2b^2+3d^2=(ad-bc)^2=6 求 :(a^2+b^2)(c^2+d^2)
由题意知:2a^2+3c^2=6①,2b^2+3d^2=6②,(ad-bc)^2=6③;①和②相乘得:4(ab)^2+6(ad)^2+6(bc)^2+9(cd)^2=36,即(2ab)^2+(3cd)^2+6[(ad)^2+(bc)^2]=36④;由③得(ad)^2+(bc)^2=6+2abcd,代入④得(2ab+3cd)^2=0,即2ab=-3cd⑤;由③得ad-bc=√6⑥或ad-bc=-√6⑦;联立⑤⑥得c=-(b√6)/3、a=3d/√6,所以ac=-bd,即ac+bd=0;联立⑤⑦得c=(b√6)/3、a=-3d/√6,所以ac=-bd,即ac+bd=0;所以ad-bc=±√6无论取正取负,ac+bd=0⑧;所以(a^2+b^2)(c^2+d^2)=(ac)^2+(ad)^2+(bc)^2+(bd)^2⑨;由③得(ad)^2+(bc)^2=6+2abcd,此式代入⑨得(a^2+b^2)(c^2+d^2)=6+(ac+bd)^2,⑧代入此式得(a^2+b^2)(c^2+d^2)=6.解毕.

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