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If two chords drawn from the point A(4,4...

If two chords drawn from the point `A(4,4)` to the parabola `x^2=4y` are bisected by the line `y=m x ,` the interval in which `m` lies is (a) `(-2sqrt(2),2sqrt(2))` (b) `(-oo,-sqrt(2))uu(sqrt(2),oo)` (c) `(-oo,-2sqrt(2)-2)uu(2sqrt(2)-2,oo)` (d) none of these

Text Solution

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Point (4,4) lies on the given parabola `x^(2)=4y`.
Any point on line y=mxis Q `(alpha,malpha)`.
Let the chord PR be bisected by Q.
`:." "alpha=(4+x_(1))/(2)rArrx_(2)=2alpha-4`
`andmalpha=(4+y_(1))/(2)rArry_(2)=2malpha-4`
Now, Q lies on the curve.
`:." "(2alpha-4)^(2)=4(2malpha-4)`
`rArr" "4alpha^(2)+16-16alpha=8(malpha-2)`
`rArr" "4alpha^(2)-8(2+m)alpha+32=0`
Since it is possible to draw two chords which get bisected at point Q, then above equation has two distinct real roots.
`:.` Discriminant, `Dgt0`
`rArr" "(8(2+m))^(2)-4xx4xx32gt0`
`rArr" "(2+m)^(2)-8gt0`
`rArr" "2+mgt2sqrt(2)or2+mgt-2sqrt(2)`
`rArr" "mgt2sqrt(2)-2ormlt-2sqrt(2)-2`
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