some high school math

today one of my friends prodded me to do some high school math. after a gap of many years i was again operating in my domain, my own turf; a long needed respite from the everyday grillings of a pgdm course. 

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