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A variable line y=m x-1 cuts the lines x...

A variable line `y=m x-1` cuts the lines `x=2y` and `y=-2x` at points `Aa n dB` . Prove that the locus of the centroid of triangle `O A B(O` being the origin) is a hyperbola passing through the origin.

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Solving the variable line `y=mx-1` with `x=2y`, we get
`x_(1)=(2)/(2m-1)" (1)"`
Solving with `y=-2x`, we get
`x_(2)=(1)/(x+2)" (2)"`
Now, `y_(1)+y_(2)=m(x_(1)+x_(2))-2`
Let the centroid of trianlge OAB be (h,k). Then,
`h=(x_(1)+x_(2))/(3)`
`"and "k=(y_(1)+y_(2))/(3)=(m(x_(1)+x_(2))-2)/(3)`
`"or "m=(3k+2)/(3h)`
So, `3h=x_(1)+x_(2)=(2)/(2((3k+2)/(3h))-1)+(1)/(((3k+2)/(3h))+2)`
`" [Using (1) and (2)]"`
`"or "(2)/(6k-3h+4)+(1)/(6h+3k+2)=1`
Simplifying, we get the final locus as `6x^(2)-9xy-6y^(2)-3x-4y=0` which is a hyperbola passing through the origin, as `h^(2)gtab and Delta ne 0`.
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