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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.


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.


Grade:12

1 Answers

mounika gajawada
31 Points
10 years ago

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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