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Find the length of normal chord which su...

Find the length of normal chord which subtends an angle of `90^0` at the vertex of the parabola `y^2=4xdot`

Text Solution

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We know that normal at point `P(at^(2),2at)` to the parabola `y^(2)=4ax` meet the parabola again at point `Q(at'^(2),2at')` such that
`t'=-t-(2)/(t)` (1)

Also, PQ subtends right angle vertex O(0,0).
`:." "tt'=-4`
From (1), `tt'=-t^(2)-2`
`rArr" "-4=-t^(2)-2`
`rArr" "t^(2)=2`
`rArr" "t=sqrt(2)` (considering point P in first quadrant)
`rArr" "t'=-2sqrt(2)`
`:." Point"P-=(2,2sqrt(2))andQ-=(8,-4sqrt(2))`
`:." "PQ=sqrt((8-2)^(2)+(-4sqrt(2)-2sqrt(2))^(2))`
`=sqrt(36+72)`
`=6sqrt(3)`
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