The points of intersection of lx+my=1 and y2=4ax are:
lx+my=1⇒my=1−lx⇒y=m1−lx
(m1−lx)2=4ax⇒m21−2lx+l2x2=4ax⇒
l2x2+(−2l−4am2)x+1=0⇒x=l2l+2am2±2ma(l+am2)
⇒y=−2am∓2a(l+am2)
The points of intersection are:
P(l2l+2am2+2ma(l+am2),−2am−2a(l+am2))=OP
Q(l2l+2am2−2ma(l+am2),−2am+2a(l+am2))=OQ
We know that the lines joining the origin O are at right angles: OP⋅OQ=0:
l2⋅l2(l+2am2+2ma(l+am2))(l+2am2−2ma(l+am2))+
+(−2am−2a(l+am2))(−2am+2a(l+am2))=
=l21−4la=0⇒l21=4la⇒1=4lal2⇒1=4l3a
We see that the problem statement is incorrect. It is correct that 4al3=1 is satisfied.
Comments