(3)设二次型f(红1,2，x)=xTAx的秩为1,4中各行元素之和为3，则∫在正交变换x=Qg下的标准形 为 (④)若二次曲面的方程x2+32+2+2y+2z+2=4经正交变换化为听+4号=4，则a= (⑤)若二次曲面的方程r2+3y2+2+2ay+2x2+2=4经正交变换化为+4=4，则a= (6)设二次型fm1,2,x3)=x2+3吃+号+2红1x2+2红1x3+2x2x3,则f的正惯性指数为- (⑦)已知二次型f(1,r2,)=片+号+c号+2x12+2红1x3经正交变换化为标准形+2，则a (⑧)二次型∫1,2，x)=(1+x22+(2-2+(c+P的秩为 00 (⑨)若实对称矩阵A与B=0 -12合同，则二次型rT4的规范形为 022 题型二，化二次型为标准形 例11.2求正交变换化二次型2x号-2x12+2x1x3-2r2江3为标准形，并写出所用正交变换 101 例11.2(2012)已知4 011 -10a 次型f(红1,2，工3)=xT(4TA)z的秩为2 0a-1 ()求实数a的值： (②)求正交变换 Qy将f化为标准形。 3 (3) g.f(x1, x2, x3) = x T Axùè1,A•à1ÉÉ⁄è3,Kf3CÜx = QyeIO/ è . (4) eg­°êßx 2+ 3y 2+z 2+ 2axy+ 2xz+ 2yz = 4²CÜzèy 2 1 + 4z 2 1 = 4,Ka = . (5) eg­°êßx 2+ 3y 2+z 2+ 2axy+ 2xz+ 2yz = 4²CÜzèy 2 1 + 4z 2 1 = 4,Ka = . (6) g.f(x1, x2, x3) = x 2 + 3x 2 2 + x 2 3 + 2x1x2 + 2x1x3 + 2x2x3,Kf.5çÍè . (7) Æg.f(x1, x2, x3) = x 2 1 + x 2 2 + cx2 3 + 2ax1x2 + 2x1x3²CÜzèIO/y 2 1 + 2y 2 3 ,Ka = . (8) g.f(x1, x2, x3) = (x1 + x2) 2 + (x2 − x3) 2 + (x3 + x1) 2ùè . (9) e¢È°› AÜB =     1 0 0 0 −1 2 0 2 2     ‹”, Kg.x T Ax5â/è . K.,zg.èIO/ ~11.2 ¶CÜzg.2x 2 3 − 2x1x2 + 2x1x3 − 2x2x3èIO/,ø—§^CÜ. ~11.2 (2012)ÆA =   1 0 1 0 1 1 −1 0 a 0 a −1   , g.f(x1, x2, x3) = x T (AT A)xùè2. (1) ¶¢Íaä; (2) ¶CÜx = QyÚfzèIO/. 3