the problem as my friend framed it : "prove that a conic section of the form ax^2+2hxy+by^2=1 is a circle, if any line drawn from a point P(x0,y0) intersects the conic section at two distinct points; A, B; such that PA.PB is constant"
ahem.. my proof
Consider a change of axis such that the point P becomes origin. The conic section w.r.t. the new co-ord sys can be represented as
ax^2+2hxy+by^2+2gx+2fy+c=0; the line can be represented as y=mx, where
PA.PB=(1+m^2)|x1*x2|; where x1 and x2 are roots of the quad:
(a+2hm+bm^2)x^2+2(g+fm)x+c=0
PA.PB=|c|(1+m^2)/|a+2hm+bm^2|=k... a constant >0
Now the expression given above should trivially hold good for all real-m
|c|(1+m^2)=k|a+2hm+bm^2|
So,
a=b=+/-|c|/k , h=0
So the conic is a circle
Ya, i agree a small one but nevertheless it was atleast worth a start, as far as my first blog is concerned
