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A parabola of latus rectum l touches a f...

A parabola of latus rectum `l` touches a fixed equal parabola. The axes of two parabolas are parallel. Then find the locus of the vertex of the moving parabola.

Text Solution

Verified by Experts

Let the fixed parabola be
`y^(2)=4ax` (1)

Since moving parabola and given parabola are equal, they have same latus rectum length.
So, latus rectum length.
`(y-k)^(2)=-4a(x-h)` (2)
On solving equation (1) and (2), we get
`(y-k)^(2)=-4a((y^(2))/(4a)-h)`
`rArr" "y^(2)-2ky+k^(2)=-y^(2)+4ah`
`rArr" "2y^(2)-2ky+k^(2)-4ah=0`
Since the two parabolas touch each other, above equation has two equal roots and therefore, discriminant is zero.
`rArr" "4k^(2)-8(k^(2)-4ah)=0`
`rArr" "-4k^(2)+32ah=0`
`rArr" "y^(2)=8ax`, which is locus of the vertex of the moving parabola.
Thus locus is parabola.
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