Question

椭圆x平方/16+y平方/4=1上有两点P.Q,O是原点，若OP.OQ斜率之积为负1/4。求证op平方+OQ平方为定值

Answer 1

设OP:y=k1xOQ:y=k2x代入原方程，得x1�0�5*（1/16+4k1�0�5/16）=1x2�0�5*（1/16+4k2�0�5/16）=1所以OP�0�5+OQ�0�5=x1�0�5+x2�0�5+y1�0�5+y2�0�5=x1�0�5*(k1�0�5+1)+x2�0�5*(k2�0�5+1)=16(8*k1�0�5*k2�0�5+5k1�0�5+5k2�0�5+2)/(4k1�0�5+1)(4k2�0�5+1�0�5)=40(2k1�0�5+2k2�0�5+1)/2(2k1�0�5+2k2�0�5+1)=20为定值