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

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