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two parabolas have a common axis and concavities in opposite directions ; if any line parallel to the common axis meets the parabolas at p and p1 prove that locus of the mid point of pp1 is another parabola provided the latus recta of the given parabolas are unequal.
for the sake of simplicity let eqn of 2 given parabolas be
y2 = 4ax........ (1)
y2 = -4a'x ......(2) { both given parabolas have unequal LLR }
now y = t be a line parallel to common axis;(where t € R )
it cuts parabola (1) at P( t2/4a , t) and (2) at P1( -t2/4a' , t )
mid pt od PP1 is ( {(a'-a)t2/8aa'} , t ) = (x , y )
now eliminatin t we get locus of midpt of PP1 as
y2 = {8aa'/(a'-a)}x
hence the locus of mid pt of PP1 is a parabola..... PROVED
plz approve if u lyk my ans
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