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Find the locus of midpoint of family of ...

Find the locus of midpoint of family of chords `lamdax+y=5(lamda` is parameter) of the parabola `x^(2)=20y`

Text Solution

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Equation of family of chord is
`(y-5)+lamda(x-0)=0`, which are concurrent at (0,5).
The given parabola is `x^(2)-20y=0`.
Let M (h,k) be midpoint of chord.
Therefore, equation of such chord is
`hx-10(y+k)=h^(2)-20k" "(Using T=S_(1))`
This chord is passing through the point (0,5).
This chord is passing through the point (0,5).
`:." "h^(2)=10k-50`
So, locus of M (h,k) is `x^(2)=10(y-5)`.
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